Chapter 5 Optimum Receivers for the Additive White Gaussian Noise Channel

Transcription

1 Chapter 5 Optimum Receiver for the Additive White Gauian Noie Channel

2 Table of Content 5.1 Optimum Receiver for Signal Corrupted by Additive White Noie Correlation Demodulator 5.1. Matched-Filter Demodulator The Optimum Detector The Maximum-Likelihood Sequence Detector A Symbol-by-Symbol MAP Detector for Signal with Memory 5. Performance of the Optimum Receiver for Memoryle Modulation 5..1 Probability of Error for Binary Modulation 5.. Probability of Error for M-ary Orthogonal Signal Fall, 4.

3 Table of Content 5..3 Probability of Error for M-ary Biorthogonal Signal 5..4 Probability of Error for Simplex Signal 5..5 Probability of Error for M-ary Binary-Coded Signal 5..6 Probability of Error for M-ary PAM 5..7 Probability of Error for M-ary PSK 5..8 Differential PSK(DPSK) and It Performance 5..9 Probability of Error for QAM 5..1 Comparion of Digital Modulation Method 5.3 Optimum Receiver for CPM Signal Optimum Demodulator and Detection of CPM 5.3. Performance of CPM Signal Fall, 4. 3

4 Table of Content Symbol-by-Symbol Detection of CPM Signal Suboptimum Demodulation and Detection of CPM Signal Optimum Receiver for Signal with Random Phae on AWGN Channel Optimum Receiver for Binary Signal 5.4. Optimum Receiver for M-ary Orthogonal Singal Probability of Error for Envelope Detection of M-ary Orthogonal Signal Probability of Error for Envelope Detection of Correlated Binary Signal Fall, 4. 4

5 5.5 Performance Analyi fir Wireline and Radio Communication Sytem Regenerative Repeater Table of Content 5.5. Link Budget Analyi in Raid Communication Sytem Fall, 4. 5

6 5.1 Optimum Receiver for Signal Corrupted by Additive White Gauian Noie We aume that the tranmitter end digital information by ue of M ignal waveform { m (t)=1,,,m }. Each waveform i tranmitted within the ymbol interval of duration T, i.e. t T. The channel i aumed to corrupt the ignal by the addition of white Gauian noie, a hown in the following figure: rt () = () t + nt (), t T where n(t) denote a ample function of AWGN proce with power pectral denity Φ nn (f )=½N W/Hz. Fall, 4. 6 m

7 5.1 Optimum Receiver for Signal Corrupted by Additive White Gauian Noie Our object i to deign a receiver that i optimum in the ene that it minimize the probability of making an error. It i convenient to ubdivide the receiver into two part the ignal demodulator and the detector. The function of the ignal demodulator i to convert the received waveform r(t) into an N-dimenional vector r=[r 1 r.. r N ] where N i the dimenion of the tranmitted ignal waveform. The function of the detector i to decide which of the M poible ignal waveform wa tranmitted baed on the vector r. Fall, 4. 7

8 5.1 Optimum Receiver for Signal Corrupted by Additive White Gauian Noie Two realization of the ignal demodulator are decribed in the next two ection: One i baed on the ue of ignal correlator. The econd i baed on the ue of matched filter. The optimum detector that follow the ignal demodulator i deigned to minimize the probability of error. Fall, 4. 8

9 5.1.1 Correlation Demodulator We decribe a correlation demodulation that decompoe the receiver ignal and the noie into N-dimenional vector. In other word, the ignal and the noie are expanded into a erie of linearly weighted orthonormal bai function {f n (t)}. It i aumed that the N bai function {f n (t)}pan the ignal pace, o every one of the poible tranmitted ignal of the et { m (t)=1 m M } can be repreented a a linear combination of {f n (t)}. In cae of the noie, the function {f n (t)} do not pan the noie pace. However we how below that the noie term that fall outide the ignal pace are irrelevant to the detection of the ignal. Fall, 4. 9

10 5.1.1 Correlation Demodulator Suppoe the receiver ignal r(t) i paed through a parallel bank of N bai function {f n (t)}, a hown in the following figure: T [ ] r() t f () t dt = () t + n() t f () t dt k m k r = + n, k = 1,,... N k mk k The ignal i now repreented by the vector m with component mk, k=1,, N. Their value depend on which of the M ignal wa tranmitted. T mk m k T T = () t f () t dt, k = 1,,...N n = n( t) f ( t) dt, k = 1,,... N k k Fall, 4. 1

11 Fall, In fact,we can expre the receiver ignal r(t) in the interval t T a: The term n'(t), defined a i a zero-mean Gauian noie proce that repreent the difference between original noie proce n(t) and the part correponding to the projection of n(t) onto the bai function {f k (t)}. = = = + = + + = N k k k N k N k k k k mk t n t f r t n t f n t f t r ) ( ) ( ) ( ) ( ) ( ) ( = = N k k k t f n t n t n 1 ) ( ) ( ) '( Correlation Demodulator

12 We hall how below that n'(t) i irrelevant to the deciion a to which ignal wa tranmitted. Conequently, the deciion may be baed entirely on the correlator output ignal and noie component r k = mk +n k, k=1,,,n. The noie component {n k } are Gauian and mean value are: E( nk) = E n() t f () for all. k t dt = n and their covariance are: Fall, 4. T Correlation Demodulator T [ ] T Autocorrelation [ ] E( nn ) = E ntn () ( τ ) f () t f ( τ) dtdτ k m k m 1 T T = N ( ) ( ) ( ) δ t τ f k t fm τ dtdτ 1 T 1 = N f ( ) ( ) k t fm t dt = Nδ mk 1 Power pectral denity i Φ nn (f)=½n W/Hz Concluion: The N noie Component {n k } are zero-mean uncorrelated Gauian random variable with a common varianceσ n =½N.

13 5.1.1 Correlation Demodulator From the above development, it follow that the correlator output {r k } conditioned on the mth ignal being tranmitted are Gauian random variable with mean E ( r ) = E( + n ) = k mk k mk and equal variance 1 σ r = σ n = N Since the noie component {n k } are uncorrelated Gauian random variable, they are alo tatitically independent. A a conequence, the correlator output {r k } conditioned on the mth ignal being tranmitted are tatitically independent Gauian variable. Fall, 4. 13

14 5.1.1 Correlation Demodulator The conditional probability denity function of the random variable r=[r 1 r r N ] are: N p( r m ) = p( rk mk ), m = 1,,..., M p( r k mk k = 1 1 ) = π N exp ( r k N mk ), k ----(A) = 1,,..., N ---(B) By ubtituting Equation (A) into Equation (B), we obtain the joint conditional PDF N 1 ( rk ) mk p( r m) = exp, 1,,..., N m= M ( π N) k = 1 N Fall, 4. 14

15 The correlator output (r 1 r r N ) are ufficient tatitic for reaching a deciion on which of the M ignal wa tranmitted, i.e., no additional relevant information can be extracted from the remaining noie proce n'(t). Indeed, n'(t) i uncorrelated with the N correlator output {r k }: N E [ n () t r ] = E[ n () t ] + E[ n () t n ] = E[ n () t n ] = E n() t n f () t n k mk k k j j k j= 1 T Correlation Demodulator [ ] = E n( t) n( τ) f ( τ) dτ E( n n ) f ( t) N k j k j j= 1 T N 1 1 = Nδ ( t τ) f ( ) k τ dτ Nδ jk f j( t) j= = Nfk() t Nfk() t = Q.E.D. Fall, T nk = n() t fk() t dt

16 5.1.1 Correlation Demodulator Since n' (t) and {r k } are Gauian and uncorrelated, they are alo tatitically independent. Conequently, n'(t) doe not contain any information that i relevant to the deciion a to which ignal waveform wa tranmitted. All the relevant information i contained in the correlator output {r k } and, hence, n'(t) can be ignored. Fall, 4. 16

17 Example Correlation Demodulator Conider an M-ary baeband PAM ignal et in which the baic pule hape g(t) i rectangular a hown in following figure. The additive noie i a zero-mean white Gauian noie proce. Let u determine the bai function f(t) and the output of the correlation-type demodulator. The energy in the rectangular pule i ε g T = g ( t) dt = a dt = a T T Fall, 4. 17

18 Example (cont.) Since the PAM ignal et ha dimenion N=1, there i only one bai function f(t) given a: 1 f ( t) = g( t) = 1 T ( t T ) a T (otherwie) The output of the correlation-type demodulator i: T 1 T r = r( t) f ( t) dt = r( t) dt T The correlator become a imple integrator when f(t) i rectangular: if we ubtitute for r(t), we obtain: 1 { T [ ] } 1 T T r= () () () () m t nt dt m tdt ntdt + = + T T = + n m Correlation Demodulator Fall, 4. 18

19 5.1.1 Correlation Demodulator Example (cont.) The noie term E(n)= and: 1 T T σn = E n() t n() t = E n() t n( τ) dtdτ T 1 T T N T T = E[ n( t) n( τ )] dtdτ = δ( t τ) dtdτ T T N T 1 = 1 dτ = N T The probability denity function for the ampled output i: 1 ( r m ) p( r m ) = exp πn N Fall, 4. 19

20 5.1. Matched-Filter Demodulator Intead of uing a bank of N correlator to generate the variable {r k }, we may ue a bank of N linear filter. To be pecific, let u uppoe that the impule repone of the N filter are: h k ( t) = f ( T t), k t T where {f k (t)} are the N bai function and h k (t)= outide of the interval t T. The output of thee filter are : Fall, 4. ( ) ( ) y () t = r t h t k t = r( τ) h ( t τ) dτ t = r( τ) f ( T t+ τ) dτ, k = 1,,..., N k k k

21 5.1. Matched-Filter Demodulator If we ample the output of the filter at t = T, we obtain T yk( T) = r( τ) f k( τ) dτ = rk, k = 1,,..., N A filter whoe impule repone h(t) = (T t), where (t) i aumed to be confined to the time interval t T, i called matched filter to the ignal (t). An example of a ignal and it matched filter are hown in the following figure. Fall, 4. 1

22 5.1. Matched-Filter Demodulator The repone of h(t) = (T t) to the ignal (t) i: t () () ( ) yt ( ) t ht ( τ ) ht τ dτ ( τ) T ( t τ) dτ = = = + which i the time-autocorrelation function of the ignal (t). Note that the autocorrelation function y(t) i an even function of t, which attain a peak at t=t. t Fall, 4.

23 5.1. Matched-Filter Demodulator Matched filter demodulator that generate the oberved variable {r k } Fall, 4. 3

24 Propertie of the matched filter. If a ignal (t) i corrupted by AWGN, the filter with an impule repone matched to (t) maximize the output ignal-to-noie ratio (SNR). Proof: let u aume the receiver ignal r(t) conit of the ignal (t) and AWGN n(t) which ha zero-mean and Φ ( f ) = 1 W/Hz. N nm Suppoe the ignal r(t) i paed through a filter with impule repone h(t), t T, and it output i ampled at time t=t. The output ignal of the filter i: t yt () = t () ht () = r( τ) ht ( τ) dτ Fall, Matched-Filter Demodulator t = ( τ ) h( t τ) dτ + n( τ) h( t τ) dτ 4 t

25 Proof: (cont.) Fall, Matched-Filter Demodulator At the ampling intant t=t: T T y( T) = ( ) h( T ) d + n( ) h( T ) d = y ( T) + y ( T) τ τ τ τ τ τ n Thi problem i to elect the filter impule repone that maximize the output SNR defined a: y ( T ) SNR = [ ( )] E yn T T T E yn ( T) = E[ n( τ) n( t) ] h( T τ) h( T t) dtdτ 1 T T 1 T = N δ( t τ) ht ( τ) ht ( tdtd ) τ = N h ( T tdt ) 5

26 5.1. Matched-Filter Demodulator Proof: (cont.) ( ) By ubtituting for y (T) and E y into SNR n T. τ ' = T τ T T ( τ ) h( T τ) dτ h( τ ') ( T τ ') dτ ' SNR = = 1 T 1 T N h ( T t) dt N h ( T t) dt Denominator of the SNR depend on the energy in h(t). The maximum output SNR over h(t) i obtained by maximizing the numerator ubject to the contraint that the denominator i held contant. Fall, 4. 6

27 5.1. Matched-Filter Demodulator Proof: (cont.) Cauchy-Schwarz inequality: if g 1 (t) and g (t) are finiteenergy ignal, then g 1( t) g( t) dt g1 ( t) dt g ( t) dt with equality when g 1 (t)=cg (t) for any arbitrary contant C. If we et g 1 (t)=h 1 (t) and g (t)=(t t), it i clear that the SNR i maximized when h(t)=c(t t). Fall, 4. 7

28 SNR 5.1. Matched-Filter Demodulator Proof: (cont.) The output (maximum) SNR obtained with the matched filter i: T T ( τ ) h( T τ) dτ ( τ) C( T ( T τ) ) dτ = = 1 T 1 T N ( ( )) ( ) N h T t dt N C T T t dt ε T = ( t) dt = N N Note that the output SNR from the matched filter depend on the energy of the waveform (t) but not on the detailed characteritic of (t). Fall, 4. 8

29 Frequency-domain interpretation of the matched filter Since h(t)=(t t), the Fourier tranform of thi relationhip i: T jπ fτ j π ft * jπ ft = ( τ) e dτ e = S ( f) e The matched filter ha a frequency repone that i the complex conjugate of the tranmitted ignal pectrum multiplied by the phae factor e -jπft (ampling delay of T). In other world, H( f ) = S( f ), o that the magnitude repone of the matched filter i identical to the tranmitted ignal pectrum. On the other hand, the phae of H( f ) i the negative of the phae of S( f ). Fall, Matched-Filter Demodulator T jπ ft ( ) = ( ) let τ = - H f T t e dt T t 9

30 Fall, Matched-Filter Demodulator Frequency-domain interpretation of the matched filter If the ignal (t) with pectrum S(f) i paed through the matched filter, the filter output ha a pectrum Y( f )= S( f ) e -jπft * jπ ft. S( f ) S ( f) e The output waveform i: () ( ) j ft π y ( ) j π ft j π ft t = Y f e df = S f e e df By ampling the output of the matched filter at t = T, we obtain (from Pareval relation): = = y ( T ) S( f ) df ( t) dt The noie at the output of the matched filter ha a power pectral denity (from equation.-7) 1 Φ ( f ) = H ( f ) N 3 T =ε

31 Frequency-domain interpretation of the matched filter The total noie power at the output of the matched filter i P n 5.1. Matched-Filter Demodulator = = Φ 1 N ( ) = 1 The output SNR i imply the ratio of the ignal power P, given by P y ( T ), to the noie power P n. = f ) df H ( f SNR = P P n df = ε N 1 ε N = S( f ε N ) df = 1 ε N Fall, 4. 31

32 5.1. Matched-Filter Demodulator Example 5.1-: M=4 biorthogonal ignal are contructed from the two orthogonal ignal hown in the following figure for tranmitting information over an AWGN channel. The noie i aumed to have a zero-mean and power pectral denity ½ N. Fall, 4. 3

33 Fall, Example 5.1-: (cont.) The M=4 biorthogonal ignal have dimenion N= (hown in figure a): The impule repone of the two matched filter are (figure b): = = = = (otherwie) ) 1 ( ) ( ) ( (otherwie) ) 1 ( ) ( ) ( 1 1 T t T t T f t h T t T T t T f t h = = (otherwie) ) 1 ( ) ( (otherwie) ) 1 ( ) ( 1 T t T T t f T t T t f 5.1. Matched-Filter Demodulator

34 Example 5.1-: (cont.) If 1 (t) i tranmitted, the repone of the two matched filter are a hown in figure c, where the ignal amplitude i A. Since y 1 (t) and y (t) are ampled at t=t, we oberve that y 1S (T)= 1, y S (T)=. From equation 5.1-7, we have ½A T=ε and the received vector i: r = [ r1 r ] = ε + n1 n where n 1 (t)=y 1n (T) and n =y n (T) are the noie component at the output of the matched filter, given by Fall, Matched-Filter Demodulator A T y kn T ( T ) = n( t) f ( t) dt, k 34 k = 1,

35 Fall, Example5.1-(cont.) Clearly, E(n k )=E[y kn (T)] and their variance i Oberve that the SNR for the firt matched filter i [ ] [ ] 1 ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( N dt t f N dtd t f f t N dtd f t f n t n E T y E T k T T k k T T k k kn n = = = = = τ τ τ δ τ τ τ σ 1 ) ( N N ε ε = = SNR 5.1. Matched-Filter Demodulator

36 5.1. Matched-Filter Demodulator Example 5.1-: (cont.) Thi reult agree with our previou reult. We can alo note that the four poible output of the two matched filter, correponding to the four poible tranmitted ignal are: ( r ) ( ) ( ) ( ) ( ) 1, r = ε + n1, n, n1, ε + n, ε + n1, n and n1, ε + n Fall, 4. 36

37 5.1.3 The Optimum Detector Our goal i to deign a ignal detector that make a deciion on the tranmitted ignal in each ignal interval baed on the obervation of the vector r in each interval uch that the probability of a correct deciion i maximized. We aume that there i no memory in ignal tranmitted in ucceive ignal interval. We conider a deciion rule baed on the computation of the poterior probabilitie defined a P( m r) P(ignal m wa tranmitted r), m=1,,,m. The deciion criterion i baed on electing the ignal correponding to the maximum of the et of poterior probabilitie { P( m r)}. Thi deciion criterion i called the maximum a poterior probability (MAP) criterion. Fall, 4. 37

38 Uing Baye rule, the poterior probabilitie may be expreed a p( r m) P( m) P( m r) = ---(A) p() r where P( m ) i the a priori probability of the mth ignal being tranmitted. The denominator of (A), which i independent of which ignal i tranmitted, may be expreed a Some implification occur in the MAP criterion when the M ignal are equally probable a priori, i.e., P( m )=1/M. The deciion rule baed on finding the ignal that maximize P( m r) i equivalent to finding the ignal that maximize P(r m ). Fall, The Optimum Detector M p r) = m= 1 ( p ( r ) P ( 38 m m )

39 The conditional PDF P(r m ) or any monotonic function of it i uually called the likelihood function. The deciion criterion baed on the maximum of P(r m ) over the M ignal i called maximum-likelihood (ML) criterion. We oberve that a detector baed on the MAP criterion and one that i baed on the ML criterion make the ame deciion a long a a priori probabilitie P( m ) are all equal. In the cae of an AWGN channel, the likelihood function p(r m ) i given by: N 1 ( rk ) mk p( r m) = exp, 1,,..., N m= M ( π N) k = 1 N contant N 1 1 = ln p( r m) N ln( πn ) ( rk mk ) N Fall, The Optimum Detector 39 k = 1

40 5.1.3 The Optimum Detector The maximum of ln p(r m ) over m i equivalent to finding the ignal m that minimize the Euclidean ditance: D( r, m ) k = 1 We called D(r, m ), m=1,,,m, the ditance metric. = N Hence, for the AWGN channel, the deciion rule baed on the ML criterion reduce to finding the ignal m that i cloet in ditance to the receiver ignal vector r. We hall refer to thi deciion rule a minimum ditance detection. ( r k mk ) Fall, 4. 4

41 5.1.3 The Optimum Detector Expanding the ditance metric: N N N m = n n mn + mn n= 1 n= 1 n= 1 D(, r ) r r = r r +, m= 1,,..., M m The term r i common to all ditance metric, and, hence, it may be ignored in the computation of the metric. The reult i a et of modified ditance metric. D ( r, m) = r m + m Note that electing the ignal m that minimize D'(r, m ) i equivalent to electing the ignal that maximize the metric C(r, m )= D'(r, m ), C r, ) = r m ( m m m Fall, 4. 41

42 The term r m repreent the projection of the ignal vector onto each of the M poible tranmitted ignal vector. The value of each of thee projection i a meaure of the correlation between the receiver vector and the mth ignal. For thi reaon, we call C(r, m ), m=1,,,m, the correlation metric for deciding which of the M ignal wa tranmitted. Finally, the term m =ε m, m=1,,,m, may be viewed a bia term that erve a compenation for ignal et that have unequal energie. If all ignal have the ame energy, m may alo be ignored. Correlation metric can be expreed a: Fall, The Optimum Detector T C(r, m) = r( t) ( ),,1,..., m t dt ε m= M m 4

43 5.1.3 The Optimum Detector Thee metric can be generated by a demodulator that crocorrelate the received ignal r(t) with each of the M poible tranmitted ignal and adjut each correlator output for the bia in the cae of unequal ignal energie. Fall, 4. 43

44 We have demontrated that the optimum ML detector compute a et of M ditance D(r, m ) or D'(r, m ) and elect the ignal correponding to the mallet (ditance) metric. Equivalently, the optimum ML detector compute a et of M correlation metric C(r, m ) and elect the ignal correponding to the larget correlation metric. The above development for the optimum detector treated the important cae in which all ignal are equal probable. In thi cae, the MAP criterion i equivalent to the ML criterion. When the ignal are not equally probable, the optimum MAP detector bae it deciion on the probabilitie given by: P( r) = m Fall, The Optimum Detector p( r ) P( ) m p() r m 44 ( r ) = ( r ) ( ) or PM, p P m m m

45 5.1.3 The Optimum Detector Example 5.1-3: Conider the cae of binary PAM ignal in which the two poible ignal point are 1 = = ε b,, where ε b i the energy per bit. The priori probabilitie are P( 1 )=p and P( )=1-p. Let u determine the metric for the optimum MAP detector when the tranmitted ignal i corrupted with AWGN. The receiver ignal vector for binary PAM i: r = ± ε b + yn (T ) where yn ( T ) i a zero mean Gauian random variable with 1 variance σ = N. n Fall, 4. 45

46 5.1.3 The Optimum Detector Example 5.1-3: (cont.) The conditional PDF P(r m ) for two ignal are 1 ( r ε ) b pr ( 1 ) = exp πσ σ n n 1 ( r + ) n pr ( ) = exp πσ n σ n Then the metric PM(r, 1 ) and PM(r, ) are p ( r ε ) b PM (, r1) = p p( r 1) = exp πσ σ n n 1 p ( r + ε b ) PM (, r) = ( 1 p) p( r ) = exp πσ σ n n Fall, ε

47 Fall, Example 5.1-3: (cont.) If PM(r, 1 ) > PM(r, ), we elect 1 a the tranmitted ignal: otherwie, we elect. Thi deciion rule may be expreed a: 1 ), ( ), ( 1 1 PM PM r r + = 1 ) ( ) ( exp 1 ), ( ), ( n b b r r p p PM PM σ ε ε r r p p r r n b b + 1 ln ) ( ) ( 1 σ ε ε p p N p p r n b = 1 ln ln 1 1 σ ε The Optimum Detector

48 5.1.3 The Optimum Detector Example 5.1-3: (cont.) 1 ln 1 p N The threhold i 4 p, denoted by τ h, divide the real line into two region, ay R 1 and R, where R 1 conit of the et of point that are greater than τ h and R conit of the et of point that are le than τ h. If r ε > τ, the deciion i made that 1 wa tranmitted. b h If r ε < τ, the deciion i made that wa tranmitted. b h Fall, 4. 48

49 5.1.3 The Optimum Detector Example 5.1-3: (cont.) The threhold τ h depend on N and p. If p=1/, τ h =. If p>1/, the ignal point 1 i more probable and, hence, τ h <. In thi cae, the region R 1 i larger than R, o that 1 i more likely to be elected than. The average probability of error i minimized It i intereting to note that in the cae of unequal priori probabilitie, it i neceary to know not only the value of the priori probabilitie but alo the value of the power pectral denity N, or equivalently, the noie-to-ignal ratio, in order to compute the threhold. When p=1/, the threhold i zero, and knowledge of N i not required by the detector. Fall, 4. 49

50 Proof of the deciion rule baed on the maximum-likelihood criterion minimize the probability of error when the M ignal are equally probable a priori. Let u denote by R m the region in the N-dimenional pace for which we decide that ignal m (t) wa tranmitted when the vector r=[r 1 r.. r N ] i received. The probability of a correct deciion given that m (t) wa tranmitted i: P( c m) p( m) = p( r m) p( m) dr The average probability of a correct deciion i: M M 1 1 Pc () = Pc ( m) = p( r m) dr m= 1M m= 1M Rm Note that P(c) i maximized by electing the ignal m if p(r m ) i larger than p(r k ) for all m k. Q.E.D. Fall, The Optimum Detector 5 R m

51 5.1.3 The Optimum Detector Similarly for the MAP criterion, when the M ignal are not equally probable, the average probability of a correct deciion i M P() c = P( r) p() r dr m= 1 R m m ame for all m. In order for P (c) to be a large a poible, the point that are to be included in each particular region R m are thoe for which P( m r) exceed all the other poterior probabilitie. Q.E.D. We conclude that MAP criterion maximize the probability of correct detection. Fall, 4. 51

52 5.1.4 The Maximum-Likelihood Sequence Detector When the ignal ha no memory, the ymbol-by-ymbol detector decribed in the preceding ection i optimum in the ene of minimizing the probability of a ymbol error. When the tranmitted ignal ha memory, i.e., the ignal tranmitted in ucceive ymbol interval are interdependent, the optimum detector i a detector that bae it deciion on obervation of a equence of received ignal over ucceive ignal interval. In thi ection, we decribe a maximum-likelihood equence detection algorithm that earche for minimum Euclidean ditance path through the trelli that characterize the memory in the tranmitted ignal. Fall, 4. 5

53 5.1.4 The Maximum-Likelihood Sequence Detector To develop the maximum-likelihood equence detection algorithm, let u conider, a an example, the NRZI ignal decribed in Section It memory i characterized by the trelli hown in the following figure: The ignal tranmitted in each ignal interval i binary PAM. Hence, there are two poible tranmitted ignal correponding to the ignal point = = ε where ε b i the energy per bit. 1 b Fall, 4. 53

54 5.1.4 The Maximum-Likelihood Sequence Detector The output of the matched-filter or correlation demodulator for binary PAM in the kth ignal interval may be expreed a r k = ± ε b + nk where n k i a zero-mean Gauian random variable with variance N. σ = n The conditional PDF for the two poible tranmitted ignal are 1 ( r ε k b ) p( rk 1) = exp πσ σ n n 1 ( r + ε ) k b p( rk ) = exp πσ σ n n Fall, 4. 54

55 Fall, Since the channel noie i aumed to be white and Gauian and f(t= it), f(t =jt) for i j are orthogonal, it follow that E(n k n j )=, k j. Hence, the noie equence n 1, n,, n k i alo white. Conequently, for any given tranmitted equence (m), the joint PDF of r 1, r,, r k may be expreed a a product of K marginal PDF, = = = = = = K k n m k k K n K k n m k k n K k m k k m k r r r p r r r p 1 ) ( 1 ) ( 1 ) ( ) ( 1 ) ( exp 1 ) ( exp 1 ) ( ),...,, ( σ πσ σ πσ (A) The Maximum-Likelihood Sequence Detector

56 5.1.4 The Maximum-Likelihood Sequence Detector Then, given the received equence r 1,r,,r k at the output of the matched filter or correlation demodulator, the detector determine the equence (m) ={ 1 (m), (m),.., K (m) } that maximize the conditional PDF p(r 1,r,,r k (m) ). Such a detector i called the maximum-likelihood (ML) equencedetector. By taking the logarithm of Equation (A) and neglecting the term that are independent of (r 1,r,,r k ), we find that an equivalent ML equence detector elect the equence (m) that minimize the Euclidean ditance metric D( r, ( m) ) = K k = 1 ( r k ( m) k ) Fall, 4. 56

57 5.1.4 The Maximum-Likelihood Sequence Detector In earching through the trelli for the equence that minimize the Euclidean ditance, it may appear that we mut compute the ditance for every poible equence. For the NRZI example, which employ binary modulation, the total number of equence i K, where K i the number of output obtained form the demodulator. However, thi i not the cae. We may reduce the number of the equence in the trelli earch by uing the Viterbi algorithm to eliminate equence a new data i received from the demodulator. The Viterbi algorithm i a equence trelli earch algorithm for performing ML equence detection. We decribe it below in the context of the NRZI ignal. We aume that the earch proce begin initially at tate S. Fall, 4. 57

58 Baic concept Viterbi Decoding Algorithm Generate the code trelli at the decoder The decoder penetrate through the code trelli level by level in earch for the tranmitted code equence At each level of the trelli, the decoder compute and compare the metric of all the partial path entering a node The decoder tore the partial path with the larger metric and eliminate all the other partial path. The tored partial path i called the urvivor. Fall, 4. 58

59 5.1.4 The Maximum-Likelihood Sequence Detector The correponding trelli i hown in the following figure : At time t=t, we receive r 1 = 1 (m) +n 1 from the demodulator, and at t=t, we receive r = (m) +n. Since the ignal memory i one bit, which we denote by L=1, we oberve that the trelli reache it regular (teady tate) form after two tranition. Differential encoding. Fall, 4. 59

60 5.1.4 The Maximum-Likelihood Sequence Detector Thu, upon receipt of r at t=t, we oberve that there are two ignal path entering each of the node and two ignal path leaving each node. The two path entering node S at t=t correpond to the information bit (,) and (1,1) or, equivalently, to the ignal point ( ε ) b, ε and ( b ε ) b, εb, repectively. The two path entering node S 1 at t=t correpond to the information bit (,1) and (1,) or, equivalently, to the ignal point ( ε ) b, ε and ( b ε ) b, εb, repectively. For the two path entering node S, we compute the two Euclidean ditance metric. D(,) = ( r1+ ε b ) + ( r + ε ) b D(1,1) = ( r1 ε b ) + ( r + ε ) b by uing the output r 1 and r from demodulator. Fall, 4. 6

61 The Viterbi algorithm compare thee two metric and dicard the path having the larger metric. The other path with the lower metric i aved and i called urvivor at t=t. Similarly, for two path entering node S 1 at t=t, we compute the two Euclidean ditance metric D1(1,) = ( r1 b ) + ( r ) b by uing the output r 1 and r from the demodulator. The two metric are compared and the ignal path with the larger metric i eliminated. Thu, at t=t, we are left with two urvivor path, one at node S and the other at node S 1, and their correponding metric. Fall, The Maximum-Likelihood Sequence Detector ε D (,1) = ( r + ) + ( r ) b 1 1 b ε 61 ε ε

62 5.1.4 The Maximum-Likelihood Sequence Detector Upon receipt of r 3 at t=3t, we compute the metric of the two path entering tate S. Suppoe the urvivor at t=t are the path (,) at S and (,1) at S 1. Then, the two metric for the path entering S at t=3t are D(,,) = D(,) + ( r3+ ε ) b D(,1,1) = D1(,1) + ( r3+ ε ) b Thee two metric are compared and the path with the larger (greater-ditance) metric i eliminated. Similarly, the metric for the two path entering S 1 at t=3t are D (,,1) = D (,) + ( r ) D1 (,1,) = D1 (,1) + ( r3 b ) Thee two metric are compared and the path with the larger metric i eliminated. Fall, ε b ε

63 5.1.4 The Maximum-Likelihood Sequence Detector It i relatively eay to generalize the trelli earch performed by the Viterbi algorithm for M-ary modulation. Delay modulation with M=4 ignal: uch ignal i characterized by the four-tate trelli hown in the following figure Fall, 4. 63

64 Fall, The Maximum-Likelihood Sequence Detector Delay modulation with M=4 ignal: (cont.) We oberve that each tate ha two ignal path entering and two ignal path leaving each node. The memory of the ignal i L=1. The Viterbi algorithm will have four urvivor at each tage and their correponding metric. Two metric correponding to the two entering path are computed at each node, and one of the two ignal path entering the node i eliminated at each tate of the trelli. The Viterbi algorithm minimize the number of trelli path earched in performing ML equence detection. From the decription of the Viterbi algorithm given above, it i unclear a to how deciion are made on the individual detected information ymbol given the urviving equence. 64

65 5.1.4 The Maximum-Likelihood Sequence Detector If we have advanced to ome tage, ay K, where K>>L in the trelli, and we compare the urviving equence, we hall find that with probability approaching one all urviving equence will be identical in bit (or ymbol) poition K-5L and le. In a practical implementation of the Viterbi algorithm, deciion on each information bit (or ymbol) are forced after a delay of 5L bit, and, hence, the urviving equence are truncated to the 5L mot recent bit (or ymbol). Thu, a variable delay in bit or ymbol detection i avoided. The lo in performance reulting from the ub-optimum detection procedure i negligible if the delay i at leat 5L. Fall, 4. 65

66 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory In thi ection, we decribe a maximum a poteriori probability algorithm that make deciion on a ymbol-by-ymbol bai, but each ymbol deciion i baed on an obervation of a equence of received ignal vector. Hence, thi detection i optimum in the ene that i minimize the probability of a ymbol error. The detection algorithm that i preented below i due to Abend and Fritchman(197),who developed it a a detection algorithm for channel with interymbol interference, i.e., channel with memory. Fall, 4. 66

67 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory We illutrate the algorithm in the context of detection a PAM ignal with M poible level. Suppoe that it i deired to detect the information ymbol tranmitted in the kth ignal interval, and let r 1,r,,r k+d be the oberved received equence, where D i the delay parameter which i choen to exceed the ignal memory, i.e., D>>L, where L i the inherent memory in the ignal. On the bai of the received equence, we compute the poterior probabilitie ( k ) P( = Am rk + D, rk + D 1,..., r1 ) for the M poible ymbol value and chooe the ymbol with the larget probability. Fall, 4. 67

68 Since P A Symbol-by-Symbol MAP Detector for Signal with Memory ( ( k ) = A r,..., r ) m k + D 1 = p( r k + D,..., r1 p( r k ) =, r k + D 1 ( Am ) P(,..., r ) and ince the denominator i common for all M probabilitie, the MAP criterion i equivalent to chooing the value of (k) that maximize the numerator of (A). Thu, the criterion for deciding on the tranmitted ymbol (k) i When the ymbol are equally probable, the probability may be dropped from the computation. ( k + D { } ( k ) ( k p( r,..., r = A ) P( A ) ~ = ) ) k + D 1 m ( k ) arg max = k ( m 1 k ) = A m ) --(A) --(B) P A m ( k ) ( = ) Fall, 4. 68

69 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory The algorithm for computing the probabilitie in Equation (B) recurively begin with the firt ymbol (1). We have ~ (1) = { } (1) (1) p( r,..., r = A ) P( = A ) arg max (1) = arg max (1) = arg max (1) 1+ D (1+ D ) 1+ D ( ) ( ) 1 p( r p 1 1+ D (,... r (1+ D) 1 m,..., (1+ D) (),...,, (1) ) (1) m ) P( (1+ D),..., (1) ) (1) where denote the deciion on (1). Fall, 4. 69

70 For mathematical convenience, we have defined P A Symbol-by-Symbol MAP Detector for Signal with Memory ( (1+ D ) () (1) ) ( (1+ D) (1) ) ( (1+ D) (1),...,, p r,..., r,..., P ) 1 1+ D 1,..., ( ) (1 + D) () (1) The joint probability P may be omitted if the 1,...,, ymbol are equally probable and tatitically independent. A a conequence of the tatitical independent of the additive noie equence, we have ( (1 + D) (1) ) 1+ D,..., 1,..., p r r = ( (1 + D) (1 + D L) ) ( ( D) ( D L) ) 1+ D,..., D,..., p r p r ( () (1) ) ( (1) ), 1 p r p r where we aume that (k) = for k Fall, 4. 7

71 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory For detection of the ymbol (),we have ~ () = = { ( ) ( )} () () p r,..., r = A P = A arg max ( ) arg max ( ) + D ( + D ) (3) 1 p ( (+ D) () ) ( (+ D) () r,..., r,..., P,..., ) + D m 1 m --(C) The joint conditional probability in the multiple ummation can be expreed a p( r + D =,..., r p 1 ( (+ D) (+ D L) ) ( (1+ D) () r,..., p r,..., r,, ) + D (+ D),..., () ) 1+ D 1 --(D) Fall, 4. 71

72 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory Furthermore, the joint probability p ( (1+ D) () ) ( (1+ D) () r,, r,, P, ) 1 + D 1, can be obtained from the probabilitie computed previouly in the detection of (1). That i, p ( (1+ D) () r ) 1+ D,, r1,, ( (1+ D) (1) ) ( (1+ D) (1) = p r,, r,, P,, ) = (1) (1) p 1 1+ D 1 ( (1+ D) () (1),,, ) --(E) Fall, 4. 7

73 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory By combining Equation (D) and (E) and then ubtituting into Equation (C),we obtain ~ () arg max ( ) = ( + D ) (3) p ( (+ D) (3) (),,, ) where, by definition, ( ( + D) (3) (),,, ) p = ( ( + D) ( + D L) ) ( ( + D) ) ( (1 + D) () (1) ) + D,, 1,,, p r P p (1) Fall, 4. 73

74 Fall, In general, the recurive algorithm for detection the ymbol (k) i a follow: upon reception of r K+D,,r,r 1, we compute where, by definition, Thu, the recurive nature of the algorithm i etablihed by relation (F) and (G). ( ) ( ) { } ( ) = = ) ( ) 1 ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( 1 ) (,,, max arg,, arg max ~ D k k k k k k D k k k k D k k p P r r p ( ) ( ) ( ) ( ) = 1) ( 1) ( ) 1 ( 1 ) ( ) ( ) ( ) ( 1) ( ) (,,,,,,, k k D k k D k L D k D k D k k k D k k p P r p p --(F) --(G) A Symbol-by-Symbol MAP Detector for Signal with Memory

75 5.1.5 A Symbol-by-Symbol MAP Detector for Signal with Memory The major problem with the algorithm i it computational complexity. In particular, the averaging performed over the ( k+ D) ( k+ 1) ( k) ymbol,,, in Equation (F) involve a large amount of computation per received ignal, epecially if the number M of amplitude level {A m } i large. On the other hand, if M i mall and the memory L i relatively hort, thi algorithm i eaily implemented. Fall, 4. 75

76 5..1 Probability of Error for Binary Modulation Let u conider binary PAM ignal where the two ignal waveform are 1 (t)=g(t) and 1 (t)= g(t), and g(t) i an arbitrary pule that i nonzero in the interval t T b and zero elewhere. Since 1 (t)= (t), thee ignal are aid to be antipodal. The energy in the pule g(t) i ε b. A indicated in ection 4.3.1, PAM ignal are one-dimenional, and, their geometric repreentation i imply the onedimenional vector = ε, = ε. 1 b b Fall, 4. Figure (A) 76

77 5..1 Probability of Error for Binary Modulation Let u aume that the two ignal are equally likely and that ignal 1 (t) wa tranmitted. Then, the received ignal from the (matched filter or correlation) demodulator i r = + n = ε + b 1 n where n repreent the additive Gauian noie component, 1 which ha zero mean and variance σ = N. In thi cae, the deciion rule baed on the correlation metric given by Equation compare r with the threhold zero. If r >, the deciion i made in favor of 1 (t), and if r<, the deciion i made that (t) wa tranmitted. n Fall, 4. 77

78 5..1 Probability of Error for Binary Modulation The two conditional PDF of r are: 1 ( r εb ) p( r 1 ) = e π N 1 ( r+ εb ) p( r ) = e π N / N / N Fall, 4. 78

79 5..1 Probability of Error for Binary Modulation Given that 1 (t) wa tranmitted, the probability of error i imply the probability that r<. ( ) 1 r ε b P( e 1) = p( r 1) dr exp = dr π N N ( r ε ) b 1 εb / N x / = e dx x= π N 1 x / = e dx ( x= x) ε / b N π ε 1 b x = Q () / Q t = N e dx t t π Fall, 4. 79

80 If we aume that (t) wa tranmitted, r = εb + n and the probability that r> i alo P( e ) ( ) = Q εb / N. Since the ignal 1 (t) and (t) are equally likely to be tranmitted, the average probability of error i 1 1 ε b Pb = P( e 1) + P( e ) = Q N --(A) Two important characteritic of thi performance meaure: Firt, we note that the probability of error depend only on the ratio ε b /N. Secondly, we note that. ε b /N i alo the output SNR from the matched-filter (and correlation) demodulator. The ratio ε b /N Fall, Probability of Error for Binary Modulation i uually called the ignal-to-noie ratio per bit. 8

81 5..1 Probability of Error for Binary Modulation We alo oberve that the probability of error may be expreed in term of the ditance between that the two ignal 1 and. From figure (A), w oberve that the two ignal are eparated by 1 the ditance d 1 = ε b. By ubtituting ε b = d1 into Equation 4 (A),we obtain d1 P = b Q N Thi expreion illutrate the dependence of the error probability on the ditance between the two ignal point. Fall, 4. 81

82 5..1 Probability of Error for Binary Modulation Error probability for binary orthogonal ignal The ignal vector 1 and are two-dimenional. 1 = [ ε ] b = [ ε ] b where ε b denote the energy for each of the waveform. Note that the ditance between thee ignal point i d = ε. 1 b Fall, 4. 8

83 5..1 Probability of Error for Binary Modulation Error probability for binary orthogonal ignal To evaluate the probability of error, let u aume that 1 wa tranmitted. Then, the received vector at the output of the demodulator i r = [ εb + n1 n]. We can now ubtitute for r into the correlation metric given by C( r, m) = r m m to obtain C(r, 1 ) and C(r, ). The probability of error i the probability that C(r, ) > C(r, 1 ). n n > n1+ εb ε ε > ε + n ε ε ( ) ( ) b b b 1 b b ( ) ( ) ( ) P e 1 = P C r > C r1 = P n n1 > ε b [,, ] [ ] Fall, 4. 83

84 5..1 Probability of Error for Binary Modulation Error probability for binary orthogonal ignal Since n 1 and n are zero-mean tatitically independent Gauian random variable each with variance ½ N, the random variable x= n n 1 i zero-mean Gauian with variance N. Hence, ( ) 1 x N P n n1 > εb = e dx ε π N b 1 x ε b = e dx= Q ε b N π N The ame error probability i obtained when we aume that i tranmitted: ε ( ) b Pb = Q = Q γb where γb i the SNR per bit. N Fall, 4. 84

85 5..1 Probability of Error for Binary Modulation If we compare the probability of error for binary antipodal ignal with that for binary orthogonal ignal, we find that orthogonal ignal required a factor of increae in energy to achieve the ame error probability a antipodal ignal. Since 1 log 1 =3 db, we ay that orthogonal ignal are 3dB poorer than antipodal ignal. The difference of 3dB i imply due to the ditance between the two ignal point, which i d = for orthogonal ignal, wherea d for antipodal ignal. 1 = 4ε b The error probability veru 1 log 1 ε b /N for thee two type of ignal i hown in the following figure (B). A oberved from thi figure, at any given error probability, the ε b /N required for orthogonal ignal i 3dB more than that for antipodal ignal. ε 1 b Fall, 4. 85

86 5..1 Probability of Error for Binary Modulation Probability of error for binary ignal Figure (B) Fall, 4. 86

87 5.. Probability of Error for M-ary Orthogonal Signal For equal-energy orthogonal ignal, the optimum detector elect the ignal reulting in the larget cro correlation between the received vector r and each of the M poible tranmitted ignal vector { m }, i.e., C To evaluate the probability of error, let u uppoe that the ignal 1 i tranmitted. Then the received ignal vector i r = εb n1 n n3 n + M where ε denote the ymbol energy and n 1,n,,n M are zeromean, mutually tatitically independent Gauian random variable with equal variance 1. Fall, 4. M ( r ) = = m r m, r, m = 1,,, M k = 1 k 87 mk σ n = N

88 5.. Probability of Error for M-ary Orthogonal Signal In thi cae, the output from the bank of M correlation are Note that the cale factor ε may be eliminated from the correlator output dividing each output by ε. With thi normalization, the PDF of the firt correlator output i Fall, 4. p r C C C ( x ) ( r, ) = ε ( ε + n ) 1 ( r, ) = ( r, M ) = ε nm 88 ε n 1 1 = exp πn 1 ( x ε ) 1 N 1

89 5.. Probability of Error for M-ary Orthogonal Signal And the PDF of the other M 1 correlator output are 1 ( ) M xm N pr xm = e, m =,3,, m πn It i mathematically convenient to firt derive the probability that the detector make a correct deciion. Thi i the probability that r 1 i large than each of the other M 1 correlator output n,n 3,, n M. Thi probability may be expreed a ( n < r1, n3 < r1,, nm r1 r1 ) p( r1 ) dr1 Pc = P < ( ) where P n < r1, n3 < r1,, nm < r1 r1 denote the joint probability that n,n 3,, n M are all le than r 1, conditioned on any given r 1. Then thi joint probability i averaged over all r 1. Fall, 4. 89

90 5.. Probability of Error for M-ary Orthogonal Signal Since the {r m } are tatitically independent, the joint probability factor into a product of M 1 marginal probabilitie of the form: P Thi probabilitie are identical for m=,3,,m, and, the joint probability under conideration i imply the reult in Equation (B) raied to the (M 1)th power. Thu, the probability of a correct deciion i M 1 1 r 1 N x Pc = e dx p( r1) dr 1 π Fall, 4. r1 ( n < r r ) = p ( x ) m 1 1 = 1 π r m r m 9 dx N m e x 1 dx m =,3,, M x = --(B) N xm

91 5.. Probability of Error for M-ary Orthogonal Signal The probability of a (k-bit) ymbol error i P =1 M P c y = r 1 N M y x 1 ε PM = 1 e dx exp y dy π π N --- (C) The ame expreion for the probability of error i obtained when any one of the other M 1 ignal i tranmitted. Since all the M ignal are equally likely, the expreion for P M given above i the average probability of a ymbol error. Thi expreion can be evaluated numerically. Fall, 4. 91

92 5.. Probability of Error for M-ary Orthogonal Signal In comparing the performance of variou digital modulation method, it i deirable to have the probability of error expreed in term of the SNR per bit, ε b /N, intead of the SNR per ymbol, ε /N. With M= k, each ymbol convey k bit of information, and hence ε = k ε b. Thu, Equation (C) may be expreed in term of ε b /N by ubtituting for ε. It i alo deirable to convert the probability of a ymbol error into an equivalent probability of a binary digit error. For equiprobable orthogonal ignal, all ymbol error are equiprobable and occur with probability PM P = M k M 1 1 Fall, 4. 9

93 5.. Probability of Error for M-ary Orthogonal Signal k Furthermore, there are way in which n bit out of k may be n in error. Hence, the average number of bit error per k-bit ymbol i k k 1 k P M n = k P k k M --(D) n= 1 n 1 1 and the average bit error probability i jut the reult in Equation (D) divided by k, the number of bit per ymbol. Thu, k 1 PM Pb = PM k >> 1 k 1 The graph of the probability of a binary digit error a a function of the SNR per bit, ε b /N, are hown in Figure (C) for M=,4,8,16,3, and 64. Thi figure illutrate that, by increaing the number M of waveform, one can reduce the SNR per bit required to achieve a given probability of a bit error. Fall, 4. 93

94 5.. Probability of Error for M-ary Orthogonal Signal For example, to achieve a P b =1 5, the required SNR per bit i a little more than 1dB for M=, but if M i increaed to 64 ignal waveform, the required SNR per bit i approximately 6dB. Thu, a aving of over 6dB i realized in tranmitter power required to achieve a P b =1 5 by increaing M from M= to M=64. Figure (C) Fall, 4. 94

95 5.. Probability of Error for M-ary Orthogonal Signal What i the minimum required ε b /N to achieve an arbitrarily mall probability of error a M? A union bound on the probability of error. Let u invetigate the effect of increaing M on the probability of error for orthogonal ignal. To implify the mathematical development, we firt derive an upper bound on the probability of a ymbol error that i much impler than the exact from given in the following Equation (5.-1) P M M y x 1 e dx ε 1 exp y π π N = dy Fall, 4. 95

96 5.. Probability of Error for M-ary Orthogonal Signal A union bound on the probability of error (cont.) Recall that the probability of error for binary orthogonal ignal i given by 5.-11: ε ( ) b Pb = Q Q b N = γ Now, if we view the detector for M orthogonal ignal a one that make M 1 binary deciion between the correlator output C(r, 1 ) that contain the ignal and the other M 1 correlator output C(r, m ), m=,3,,m, the probability of error i upper-bounded by union bound of the M 1 event. That i, if E i repreent the event that C(r, i )> C(r, 1 ) for i 1, M M then we have. Hence, ( ) ( ) i= P P E P E = M i i i= ( 1) ( 1 ) ( ε / ) ( ) ε / P M P = M Q N < MQ N M b S S Fall, 4. 96

97 5.. Probability of Error for M-ary Orthogonal Signal A union bound on the probability of error(cont.) Thi bound can be implified further by upper-bounding We have thu, A k or equivalently, a M, the probability of error approache zero exponentially, provided that ε b /N i greater than ln, ε b > ln = 1.39 ( 1.4dB) N Fall, 4. Q ( ε / ) S N ε P < Me = e P M M Q < ( ) ε N ε N < e e k N k kε N ( ε N ) b 97 b ln --(E) --(F)

98 5.. Probability of Error for M-ary Orthogonal Signal A union bound on the probability of error(cont.) The imple upper bound on the probability of error given by Equation (F) implie that, a long a SNR>1.4 db, we can achieve an arbitrarily low P M.. However, thi union bound i not a very tight upper bound a a ufficiently low SNR due to the fact that upper bound for the Q function in Equation (E) i looe. In fact, by more elaborate bounding technique, it i hown in Chapter 7 that the upper bound in Equation (F) i ufficiently tight for ε b /N >4 ln. For ε b /N <4 ln, a tighter upper bound on P M i P M < e k ( ε N ln ) Fall, 4. 98

99 5.. Probability of Error for M-ary Orthogonal Signal A union bound on the probability of error(cont.) Conequently, P M a k, provided that ε b N > ln =.693 ( 1.6dB) Hence, 1.6 db i the minimum required SNR per bit to achieve an arbitrarily mall probability of error in the limit a k (M ). Thi minimum SNR per bit i called the Shannon limit for an additive Gauian noie channel. Fall, 4. 99

100 Fall, Probability of Error for M-ary Biorthogonal Signal A indicated in Section 4.3, a et of M= k biorthogonal ignal are contructed from ½ M orthogonal ignal by including the negative of the orthogonal ignal. Thu, we achieve a reduction in the complexity of the demodulator for the biorthogonal ignal relative to that for orthogonal ignal, ince the former i implemented with ½ M cro correlation or matched filter, wherea the latter required M matched filter or cro correlator. Let u aume that the ignal 1 (t) correponding to the vector 1 =[ ε ] wa tranmitted. The received ignal vector i r = [ ε + n1 n nm ] where the {n m } are zero-mean, mutually tatitically independent and identically ditributed Gauian random variable with 1 variance σ = N. n 1

101 5..3 Probability of Error for M-ary Biorthogonal Signal The optimum detector decide in favor of the ignal correponding to the larget in magnitude of the cro correlator ( ) M C r, m = r m = r k mk, m= 1,,..., M k = 1 while the ign of thi larget term i ued to decide whether m (t) or m (t) wa tranmitted. According to thi deciion rule, the probability of a correct deciion i equal to the probability that r and r 1 = ε + n1 > 1 exceed r m = n m for m=,3, ½ M. But 1 1 ( ) r1 r1 N x N y P nm < r1 r1 > = e dx= e dy r1 r1 N πn π 1 y = N x Fall, 4. 11

102 5..3 Probability of Error for M-ary Biorthogonal Signal Then, the probability of a correct deciion i P 1 r1 N ( ) e x c = p r dr r N π from which, upon ubtitution for p(r 1 ), we obtain P 1 1 v+ ε N x v c ( N v N ) e dx e = dv ε + ε π π 1 x 1 ε p ( ) v= r1 ε r x1 = exp 1 πn N where we have ued the PDF of r 1 given in Equation The probability of a ymbol error P M =1 P c M M 1 --(G) ( ) N ( ) Fall, 4. 1

103 5..3 Probability of Error for M-ary Biorthogonal Signal P c, and P M may be evaluated numerically for different value of M. The graph hown in the following figure (D) illutrate P M a a function of ε b /N, where ε =k ε b, for M=,4,8,16 and 3. Fall, 4. 13

104 5..3 Probability of Error for M-ary Biorthogonal Signal In thi cae, the probability of error for M=4 i greater than that for M=. Thi i due to the fact that we have plotted the ymbol error probability P M in Figure(D). If we plotted the equivalent bit error probability, we hould find that the graph for M= and M=4 coincide. A in the cae of orthogonal ignal, a M (k ), the minimum required ε b /N to achieve an arbitrarily mall probability of error i 1.6 db, the Shannon limit. Fall, 4. 14

105 Fall, Probability of Error for Simplex Signal Next we conider the probability of error for M implex ignal. Recall from Section 4.3 that implex ignal are a et of M equally correlated with mutual cro-correlation coefficient ρ mn = 1/(M 1). Thee ignal have the ame minimum eparation of ε between adjacent ignal point in M-dimenional pace a orthogonal ignal. They achieve thi mutual eparation with a tranmitted energy of ε (M 1)/M, which i le than that required for orthogonal ignal by a factor of (M 1)/M. Conequently, the probability of error for implex ignal i identical to the probability of error for orthogonal ignal, but thi performance i achieved with aving of M 1log( 1 ρ ) = 1log db M 1 For M=, the aving i 3dB. A M i increaed, the aving in SNR approache db. 15

106 5..5 Probability of Error for M-ary Binary-Coded Signal In Section 4.3, we have hown that binary-coded ignal waveform are repreented by ignal vector (4.3-38) : where mj = ± N for all m and j. N i the block length of the code and i alo the dimenion of the M ignal waveform ( e) If d i the minimum Euclidean ditance, then the probability min of a ymbol error i upper-bounded a ( ) ( ) ( ) ( e ) dmin P M < M 1 Pb = M 1 Q N ( ( e) ) < k d min exp 4N Fall, 4. [ ], m = 1,, M m = m1 m mn, ε 16

107 5..6 Probability of Error for M-ary PAM Recall (4.3-6) that M-ary PAM ignal are repreent geometrically a M one-dimenional ignal point with value : 1 = ε A m= 1,,, M m g m ( ) M where A m = m 1 M d, m = 1,,, The Euclidean ditance between adjacent ignal point i d ε g. Auming equally probable ignal, the average energy i : M M 1 1 d ε M ε g av = εm = m = ( m 1 M) M m= 1 M m= 1 M m= 1 d ε M M g 1 ( ) 1 ( M + 1) ( ) = M M 1 = M 1 dε m = g m= 1 M 3 6 M M ( M + 1)( M + 1) m = m= 1 6 Fall, 4. 17

108 5..6 Probability of Error for M-ary PAM Equivalently, we may characterize thee ignal in term of their average power, where i : ε ( ) d av 1 P M ε g av = = 1 (5.-4) T 6 T The average probability of error for M-ary PAM : The detector compare the demodulator output r with a et of M-1 threhold, which are placed at the midpoint of ucceive amplitude level and deciion i made in favor of the amplitude level that i cloe to r. d 1 ε g We note that if the mth amplitude level i tranmitted, the demodulator output i 1 r = m + n= ε g Am + n where the noie variable n ha zero-mean and variance σ = 1 n N Fall, 4. 18

109 5..6 Probability of Error for M-ary PAM Auming all amplitude level are equally likely a priori, the average probability of a ymbol error i the probability that the noie variable n exceed in magnitude one-half of the ditance between level. However, when either one of the two outide level ±(M-1) i tranmitted, an error can occur in one direction only. A a reult, we have the error probability: M 1 1 ε M 1 M M π N M 1 y = e dy where y x N M d g N π = ε 1 x () / Qt = e dx t ( M 1) d ε t π g = Q M N x N PM = P r m > d g = e dx d ε g Fall, i( M ) + 1 i 1 i M M

110 5..6 Probability of Error for M-ary PAM From (5.-4), we note that 6 P M 1 By ubtituting for d ε g, we obtain the average probability of a ymbol error for PAM in term of the average power : ( M 1) 6PT ( M 1) 6ε av av PM = Q = Q M ( M 1) N M ( ) M 1 N It i cutomary for u to ue the SNR per bit a the baic parameter, and ince T=kT b and k =log M: P M = where ε bav = PT av b i the average bit energy and εb av SNR per bit. d ε g = ( M 1) ( 6log M ) M Q ε b av ( ) M 1 N av T No i the average Fall, 4. 11

111 5..6 Probability of Error for M-ary PAM The cae M= correpond to the error probability for binary antipodal ignal. The SNR per bit increae by over 4 db for every factor-of- increae in M. Probability of a ymbol error for PAM Fall, For large M, the additional SNR per bit required to increae M by a factor of approache 6 db.

112 5..7 Probability of Error for M-ary PSK Recall from ection 4.3 that digital phae-modulated ignal waveform may be expreed a (4.3-11): π m t = g t co πf ct + m 1 M, 1 m M, and have the vector repreentation: () () ( ) t T m t ε π m ε π m ε 1 = co 1 in 1 M M, = Since the ignal waveform have equal energy, the optimum detector for the AWGN channel given by Equation compute the correlation metric () ( ) ( ) ε g C ( r, ) = r, m = 1,, M m m, Energy in each waveform. (from 4.3-1) Fall, 4. 11

113 5..7 Probability of Error for M-ary PSK In other word, the received ignal vector r=[r 1 r ] i projected onto each of the M poible ignal vector and a deciion i made in favor of the ignal with the larget projection. Thi correlation detector i equivalent to a phae detector that compute the phae of the received ignal from r. We elect the ignal vector m whoe phae i cloer to r. The phae of r i 1 r Θr = tan r 1 We will determine the PDF of Θ r, and compute the probability of error from it. Conider the cae in which the tranmitted ignal phae i =, correponding to the ignal 1 (t). Θ r Fall,

114 5..7 Probability of Error for M-ary PSK The tranmitted ignal vector i 1 = ε, and the received ignal vector ha component: r1 = ε + n1 r = n Becaue n 1 and n are jointly Gauian random variable, it follow that r 1 and r are jointly Gauian random variable variable with 1 E( r 1 ) = ε, Er ( ) =, and σ r = σ 1 r = N = σr. ( ) 1 r ε 1 + r p ( ) r r1, r = exp πσ r σ r The PDF of the phaeθ r i obtained by a change in variable from (r 1,r ) to : 1 r V = r1 + r Θ r = tan r Fall,

115 The joint PDF of V and Θ r : p, ( ) p Θ ( Θ ) r r ( ) ( ) Integration of pv Θ V, Θ r r over the range of V yield p Θ = p V, Θ dv Θ r 5..7 Probability of Error for M-ary PSK 1 γ ( co ) / in Θ V γ Θr σ r r = e Ve dv πσ r where we define the ymbol SNR a γ = ε N. Fall, 4. V exp V V, Θ ( V, Θ ) = r r πσ r σ r r V, Θr r = 1 πσ ε ε V coθ V N co r r N in V ε Θ σ ε Θr N e e dv r N dr1dr = VdVdΘ r V = V N

116 5..7 Probability of Error for M-ary PSK fθ ( Θ ) become r r narrower and more peaked about Θ r = a the SNR increae. γ Probability denity function p ( Θ ) for 1,, 4, and 1. r r γ = Θ S Fall,

117 5..7 Probability of Error for M-ary PSK When 1 (t) i tranmitted, a deciion error i made if the noie caue the phae to fall outide the range π M Θr π M. Hence, the probability of a ymbol error i In general, the integral of pθ Θ doen t reduced to a imple r r form and mut be evaluated numerically, except for M = and M =4. For binary phae modulation, the two ignal 1 (t) and (t) are antipodal. Hence, the error probability i Fall, 4. M π M π M 117 ( Θr ) d r P = 1 pθ Θ r P = Q ( ) ε b N

118 5..7 Probability of Error for M-ary PSK When M = 4, we have in effect two binary phae-modulation ignal in phae quadrature. Since there i no crotalk or interference between the ignal on the two quadrature carrier, the bit error probability i identical to that of M =. (5.-57) Then the ymbol error probability for M=4 i determined by noting that ( ) ε 1 1 b P c = P = Q N where P c i the probability of a correct deciion for the -bit ymbol. There, the ymbol error probability for M = 4 i ε 1 ε b b P4 = 1 Pc = Q 1 Q N N Fall,

119 5..7 Probability of Error for M-ary PSK For M > 4, the ymbol error probability P M i obtained by π M numerically integrating Equation PM = 1 pθ ( Θr ) dθr. r π M For large value of M, doubling the number of phae require an additional 6 db/bit to achieve the ame performance Fall,

120 5..7 Probability of Error for M-ary PSK An approximation to the error probability for large M and for large SNR may be obtained by firt approximating p ( Θ). For ε Θr N >> 1 and Θr.5π, p Θr ( Θ) i well approximated a : γ γ Θr PΘ ( Θ ) e r r coθ in r π Performing the change in variable from Θ, r to u = ν in Θr P M 1 π = Q π M π M γ in γ γ coθ π ( π M ) in e π M u r e du γ in = Q Θ r dθ kγ b r in k log ν = k π M = ν b M Fall, 4. 1

121 5..7 Probability of Error for M-ary PSK When a Gray Code i ued in the mapping of k-bit ymbol into the correponding ignal phae, two k-bit ymbol correponding to adjacent ignal phae differ in only a ignal bit. The mot probable error reult in the erroneou election of an adjacent phae to the true one. Mot k-bit ymbol error contain only a ingle-bit error. The equivalent bit error probability for M-ary PSK i well approximated a : 1 Pb P M k In practice, the carrier phae i extracted from the received ignal by performing ome nonlinear operation that introduce a phae ambiguity. Fall, 4. 11

122 5..7 Probability of Error for M-ary PSK For binary PSK, the ignal i often quared in order to remove the modulation, and the double-frequency component that i generated i filtered and divided by in frequency in order to extract an etimate of the carrier frequency and phae φ. Thi operation reult in a phae ambiguity of 18 in the carrier phae. For four-phae PSK, the received ignal i raied to the fourth power to remove the digital modulation, and the reulting fourth harmonic of the carrier frequency i filtered and divided by 4 in order to extract the carrier component. Thee operation yield a carrier frequency component containing φ, but there are phae ambiguitie of ±9 and 18 in the phae etimate. Conequently, we do not have an abolute etimate of the carrier phae for demodulation. Fall, 4. 1

123 5..7 Probability of Error for M-ary PSK The phae ambiguity problem can be overcome by encoding the information in phae difference between ucceive ignal tranmiion a oppoed to abolute phae encoding. For example, in binary PSK, the information bit 1 may be tranmitted by hifting the phae of carrier by 18 relative to the previou carrier phae. Bit i tranmitted by a zero phae hift relative to the phae in the previou ignaling interval. In four-phae PSK, the relative phae hift between ucceive interval are, 9, 18, and -9,correponding to the information bit, 1, 11, and 1, repectively. The PSK ignal reulting from the encoding proce are aid to be differentially encoded. Fall, 4. 13

124 5..7 Probability of Error for M-ary PSK The detector i relatively imple phae comparator that compare the phae of the demodulated ignal over two conecutive interval to extract the information. Coherent demodulation of differentially encoded PSK reult in a higher probability of error than that derived for abolute phae encoding. With differentially encoded PSK, an error in the demodulated phae of the ignal in any given interval will uually reult in decoding error over two conecutive ignaling interval. The probability of error in differentially encoded M-ary PSK i approximately twice the probability of error for M-ary PSK with abolute phae encoding. Fall, 4. 14

125 5..8 Differential PSK (DPSK) and It Performance The received ignal of a differentially encoded phae-modulated ignal in any given ignaling interval i compared to the phae of the received ignal from the preceding ignaling interval. We demodulate the differentially encoded ignal by multiplying r(t) by coπf c t and inπf c t integrating the two product over the interval T. At the kth ignaling interval, the demodulator output : r or equivalently, where θ k i the phae angle of the tranmitted ignal at the kth ignaling interval, φ i the carrier phae, and n k =n k1 +jn k i the noie vector. Fall, 4. ε ( θ φ) n ( θ φ) = co + in + n k k k k k ε 15 ε 1 ( θ φ ) j k rk = e + n k

126 Similarly, the received ignal vector at the output of the demodulator in the preceding ignaling interval i : j θ k 1 φ rk 1 = e + nk 1 The deciion variable for the phae detector i the phae difference between thee two complex number. Equivalently, we can project r k onto r k-1 and ue the phae of the reulting complex number : r k r which, in the abence of noie, yield the phae differenceθ k - θ k-1. Differentially encoded PSK ignaling that i demodulated and detected a decribed above i called differential PSK (DPSK). Fall, Differential PSK (DPSK) and It Performance ε ε ε 16 ( ) ( θ θ ) j( θ φ ) * j( θ φ ) * * j k k 1 k k 1 k 1 = e + e nk 1 + e nk + ε n k n k 1

127 5..8 Differential PSK (DPSK) and It Performance If the pule g(t) i rectangular, the matched filter may be replaced by integrate-and-dump filter Block diagram of DPSK demodulator Fall, 4. 17

128 5..8 Differential PSK (DPSK) and It Performance The error probability performance of a DPSK demodulator and detector The derivation of the exact value of the probability of error for M-ary DPSK i extremely difficult, except for M =. Without lo of generality, uppoe the phae difference θ k -θ k-1 =. Furthermore, the exponential factor e -j(θ k-1 -φ) and e j(θ k -φ) can be aborbed into Gauian noie component n k-1 and n k, without changing their tatitical propertie. r k r ε ε ( * ) * n + n n n * k 1 = + k k 1 + k k 1 The complication in determining the PDF of the phae i the term n k n* k-1. Fall, 4. 18

129 5..8 Differential PSK (DPSK) and It Performance However, at SNR of practical interet, the term n k n* k-1 i mall relative to the dominant noie term (n k +n* k-1 ). We neglect the term n k n* k-1 and normalize r k r* k-1 by dividing through by, the new et of deciion metric become : ε ε ( ) * x = + Re nk + nk 1 ( * y = Im n ) k + nk 1 The variable x and y are uncorrelated Gauian random variable with identical variance σ n =N. The phae i y Θr = tan 1 x The noie variance i now twice a large a in the cae of PSK. Thu we can conclude that the performance of DPSK i 3 db poorer than that for PSK. ε Fall, 4. 19

130 5..8 Differential PSK (DPSK) and It Performance Thi reult i relatively good for M 4, but it i peimitic for M = that the lo in binary DPSK relative to binary PSK i le than 3 db at large SNR. In binary DPSK, the two poible tranmitted phae difference are and π rad. Conequently, only the real part of r k r* k-1 i need for recovering the information. ( * ) 1 ( * * Re r ) krk 1 = rk rk 1 + rk rk 1 Becaue the phae difference the two ucceive ignaling interval i zero, an error i made if Re(r k r* k-1 )<. The probability that r k r* k-1 +r* k r k-1 < i a pecial cae of a derivation, given in Appendix B. Fall, 4. 13

131 5..8 Differential PSK (DPSK) and It Performance Appendix B concerned with the probability that a general quadratic form in complex-valued Gauian random Variable i le than zero. According to Equation B-1, we find it depend entirely on the firt and econd moment of the complex-valued Gauian random variable r k and r k-1. We obtain the probability of error for binary DPSK in the form where ε b /N i the SNR per bit. P b = 1 N e ε b Fall,

132 5..8 Differential PSK (DPSK) and It Performance The probability of a binary digit error for four-phae DPSK with Gray coding can be expre in term of well-known function, but it derivation i quite involved. According to Appendix C, it i expreed in the form : P b = Q 1 1 ( ) ( ) ( ) a, b I ab exp a + b where Q 1 (a,b) i the Marcum Q function (.1-1), I (x) i the modified Beel function of order zero(.1-13), and the parameter a and b are defined a 1 a 1 1 = γ = b 1,and b γ b 1 + Fall, 4. 13

133 5..8 Differential PSK (DPSK) and It Performance Becaue binary DPSK i only lightly inferior to binary PSK at large SNR, and DPSK doe not require an elaborate method for etimate the carrier phae, it i often ued in digital communication ytem. Probability of bit error for binary and four-phae PSK and DPSK Fall,

134 5..9 Probability of Error for QAM Recall from Section 4.3 that QAM ignal waveform may be expreed a (4.3-19) m ( t) = A g( t)co πf t A ( t)in πf where A mc and A m are the information-bearing ignal amplitude of the quadrature carrier and g(t) i the ignal pule. The vector repreentation of thee waveform i m mc A 1 ε = mc g Am g To determine the probability of error for QAM, we mut pecify the ignal point contellation. c m 1 ε c t Fall,

135 5..9 Probability of Error for QAM QAM ignal et that have M = 4 point. Figure (a) i a four-phae modulated ignal and Figure (b) i with two amplitude level, labeled A 1 and A, and four phae. Becaue the probability of error i dominated by the minimum ditance between pair of ignal point, let u impoe the condition ( that d e ) and we evaluate the average tranmitter power, baed on the premie = min A that all ignal point are equally probable. Fall,

136 5..9 Probability of Error for QAM For the four-phae ignal, we have 1 P av = ( 4) A = A 4 For the two-amplitude, four-phae QAM, we place the point ( on circle of radii A and 3A. Thu, d e ), and min = A 1 [ ( ) ] P av = 3A + A = A which i the ame average power a the M = 4-phae ignal contellation. Hence, for all practical purpoe, the error rate performance of the two ignal et i the ame. There i no advantage of the two-amplitude QAM ignal et over M = 4-phae modulation. Fall,

137 5..9 Probability of Error for QAM QAM ignal et that have M = 8 point. We conider the four ignal contellation : Auming that the ignal point are equally probable, the average tranmitted ignal power i : P av = = 1 M A M M m= 1 M ( A + A ) ( a + ) mc am m= 1 mc m The coordinate (A mc,a m ) for each ignal point are normalized by A Fall,

138 5..9 Probability of Error for QAM The two et (a) and (c) contain ignal point that fall on a rectangular grid and have P av = 6A. The ignal et (b) require an average tranmitted power P av = 6.83A, and (d) require P av = 4.73A. The fourth ignal et (d) require approximately 1 db le power than the firt two and 1.6 db le power than the third to achieve the ame probability of error. The fourth ignal contellation i known to be the bet eightpoint QAM contellation becaue it require the leat power for a given minimum ditance between ignal point. Fall,

139 5..9 Probability of Error for QAM QAM ignal et for M 16 For 16-QAM, the ignal point at a given amplitude level are phae-rotated by relative to the ignal point at adjacent amplitude level. However, the circular 16-QAM contellation i not the bet 16-point QAM ignal contellation for the AWGN channel. Rectangular M-ary QAM ignal are mot frequently ued in practice. The reaon are : Rectangular QAM ignal contellation have the ditinct advantage of being eaily generated a two PAM ignal impreed on phae-quadrature carrier. Fall,

140 The average tranmitted power required to achieve a given minimum ditance i only lightly greater than that of the bet M-ary QAM ignal contellation. For rectangular ignal contellation in which M = k, where k i even, the QAM ignal contellation i equivalent to two PAM ignal on quadrature carrier, each having M = k ignal point. The probability of error for QAM i eaily determined from the probability of error for PAM. Specifically, the probability of a correct deciion for the M-ary QAM ytem i P = P where Fall, Probability of Error for QAM p M i the probability of c ( ) 1 M 14 error of an M ary PAM.

141 5..9 Probability of Error for QAM By appropriately modifying the probability of error for M-ary PAM, we obtain 1 3 = ε av P 1 Q M M M 1 N where ε av /N o i the average SNR per ymbol. Therefore, the probability of a ymbol error for M-ary QAM i P M ( 1 P ) 1 M Note that thi reult i exact for M = k when k i even. = When k i odd, there i no equivalent M -ary PAM ytem. There i till ok, becaue it i rather eay to determine the error rate for a rectangular ignal et. Fall,

142 5..9 Probability of Error for QAM We employ the optimum detector that bae it deciion on the optimum ditance metric (5.1-43), it i relatively traightforward to how that the ymbol error probability i tightly upper-bounded a P M 1 Q 4Q 3ε av ( M 1) 3kε ( ) bav M 1 N for any k 1, whereε bav /N i the average SNR per bit. N Fall, 4. 14

143 5..9 Probability of Error for QAM For nonrectangular QAM ignal contellation, we may upperbound the error probability by ue of a union bound : ( ) [ ( )] e P < M M 1 Q dmin N ( e) where d min i the minimum Euclidean ditance between ignal point. Thi bound may be looe when M i large. We approximate P M by replacing M-1 by M n, where M n i the ( e) larget number of neighboring point that are at ditance dmin from any contellation point. It i intereting to compare the performance of QAM with that of PSK for any given ignal ize M. Fall,

144 5..9 Probability of Error for QAM For M-ary PSK, the probability of a ymbol error i approximate a π PM Q γ in M where γ i the SNR per ymbol. For M-ary QAM, the error probability i : 1 3 = ε av P 1 Q M M M 1 N We imply compare the argument of Q function for the two ignal format. The ratio of thee two argument i R M = in ( M ) ( π M ) 3 1 Fall,

145 5..9 Probability of Error for QAM We can find out that when M = 4, we have R M = 1. 4-PSK and 4-QAM yield comparable performance for the ame SNR per ymbol. For M>4, M-ary QAM yield better performance than M-ary PSK. Fall,

146 5..1 Comparion of Digital Modulation Method One can compare the digital modulation method on the bai of the SNR required to achieve a pecified probability of error. However, uch a comparion would not be very meaningful, unle it were made on the bai of ome contraint, uch a a fixed data rate of tranmiion or, on the bai of a fixed bandwidth. For multiphae ignal, the channel bandwidth required i imply the bandwidth of the equivalent low-pa ignal pule g(t) with duration T and bandwidth W, which i approximately equal to the reciprocal of T. R Since T=k/R=(log M)/R, it follow that W = log M Fall,

147 5..1 Comparion of Digital Modulation Method A M i increaed, the channel bandwidth required, when the bit rate R i fixed, decreae. The bandwidth efficiency i meaured by the bit rate to bandwidth ratio, which i R log M W = The bandwidth-efficient for tranmitting PAM i ingle-ideband. The channel bandwidth required to tranmit the ignal i approximately equal to 1/T and, R log M W = thi i a factor of better than PSK. For QAM, we have two orthogonal carrier, with each carrier having a PAM ignal. Fall,

148 Thu, we double the rate relative to PAM. However, the QAM ignal mut be tranmitted via double-ideband. Conequently, QAM and PAM have the ame bandwidth efficiency when the bandwidth i referenced to the band-pa ignal. A for orthogonal ignal, if the M = k orthogonal ignal are contructed by mean of orthogonal carrier with minimum frequency eparation of 1/T, the bandwidth required for tranmiion of k = log M information bit i M M M W = = = R T ( k R) log M In the cae, the bandwidth increae a M increae. In the cae of biorthogonal ignal, the required bandwidth i one-half of that for orthogonal ignal. Fall, Comparion of Digital Modulation Method 148

149 5..1 Comparion of Digital Modulation Method A compact and meaningful comparion of modulation method i one baed on the normalized data rate R/W (bit per econd per hertz of bandwidth) veru the SNR per bit (ε b /N ) required to achieve a given error probability. In the cae of PAM, QAM, and PSK, increaing M reult in a higher bit-to-bandwidth ratio R/W. Fall,

150 5..1 Comparion of Digital Modulation Method However, the cot of achieving the higher data rate i an increae in the SNR per bit. Conequently, thee modulation method are appropriate for communication channel that are bandwidth limited, where we deire a R/W >1 and where there i ufficiently high SNR to upport increae in M. Telephone channel and digital microwave ratio channel are example of uch band-limited channel. In contrat, M-ary orthogonal ignal yield a R/W 1. A M increae, R/W decreae due to an increae in the required channel bandwidth. The SNR per bit required to achieve a given error probability decreae a M increae. Fall, 4. 15

151 5..1 Comparion of Digital Modulation Method Conequently, M-ary orthogonal ignal are appropriate for power-limited channel that have ufficiently large bandwidth to accommodate a large number of ignal. A M, the error probability can be made a mall a deired, provided that SNR>.693 (-1.6dB). Thi i the minimum SNR per bit required to achieve reliable tranmiion in the limit a the channel bandwidth W and the correponding R/W. The figure above alo hown the normalized capacity of the band-limited AWGN channel, which i due to Shannon (1948). The ratio C/W, where C (=R) i the capacity in bit/, repreent the highet achievable bit rate-to-bandwidth ratio on thi channel. Hence, it erve the upper bound on the bandwidth efficiency of any type of modulation. Fall,

152 5.3 Optimum Receiver for CPM Signal CPM i a modulation method with memory. The memory reult from the continuity of the tranmitted carrier phae from one ignal interval to the next. The tranmitted CPM ignal may be expreed a t = co πf ct + φ t T where φ(t ;I) i the carrier phae. ε () [ ( ;I)] The filtered received ignal for an additive Gauian noie channel i r t = t + n t ( ) ( ) ( ) where n () t = n ( t) co πf t n ( t)in πf t c c c Fall, 4. 15

153 Fall, Optimum Demodulation and Detection of CPM The optimum receiver for thi ignal conit of a correlator followed by a maximum-likelihood equence detector that earche the path through the tate trelli for the minimum Euclidean ditance path. And the Viterbi algorithm i an efficient method for performing thi earch. The carrier phae for a CPM ignal with a fixed modulation index h may be expreed a φ ( t ; I) = πh I kq( t kt ) = πh = θ n k = n L k n I k + πh I kq( t kt ) = k = n L+ 1 θ ( t ; I) nt t ( n + 1)T q() t g( τ ) + where q(t)= for t <, q(t)=1/ for t LT,and 153 n = t dτ

154 5.3.1 Optimum Demodulation and Detection of CPM The ignal pule g(t) = for t < and t LT. For L = 1, we have a full repone CPM, and for L > 1, where L i a poitive integer, we have a partial repone CPM ignal. h i rational, i.e., h = m/p where m and p are relatively prime poitive integer, the CPM cheme can be repreented by a trelli. There are p phae tate when m i even (4.3-6) : ( ) πm πm p 1 πm Θ =,,,, p p p There are p phae tate when m i odd (4.3-61) : Θ ( p ) πm 1 πm =,,, p p Fall,

155 Phae Tree Thee phae diagram are called phae tree. CPFSK with binary ymbol I n =±1, the et of phae trajectorie beginning at time t=. Fall, Phae trajectorie for quaternary CPFSK.

156 5.3.1 Optimum Demodulation and Detection of CPM We know if L > 1, we have an additional number of tate due to the partial repone character of the ignal g(t). Thee additional tate can be identified by expreing θ(t ; I) : n 1 ( t;i) = h I q( t kt) + hi q( t nt) θ π π k= n L+ 1 k The firt term on the right-hand ide depend on the information ymbol (I n-1, I n-,, I n-l+1 ), which i called the correlative tate vector, and repreent the phae term correponding to ignal pule that have not reached their final value. The econd term repreent the phae contribution due to the mot recent ymbol I n. n Fall,

157 Hence, the tate of the CPM ignal (or the modulator) at time t = nt may be expreed a the combined phae tate and correlative tate, denoted a S n = { θ n, I n 1, I n,, I n L+ 1 } for a partial repone ignal pule of length LT, where L > 1. In thi cae, the number of tate i M i the alphabet ize L 1 pm ( even m) N = L 1 pm ( odd m) where h=m/p. Suppoe the tate of the modulator at t = nt i S n. The effect of the new ymbol in the time interval nt t (n+1)t i to change the tate from S n to S n+1. At t = (n+1)t, the tate where θ Optimum Demodulation and Detection of CPM ( θ I, I, I ) S n+ 1 = n+ 1, n n 1, n L+ n+ 1 = θn + πhin L+1 Fall,

158 Example A binary CPM cheme with a modulation index h = ¾ and a partial repone pule with L =. Determine the tate S n of the CPM cheme and ketch the phae tree and tate trelli. Firt, we note that there are p = 8 phae tate, namely, For each of thee phae tate, there are two tate that reult from the memory of the CPM cheme. Hence, the total number of tate i N = 16, namely, π π π π 3π (,1),(, 1),( π,1),( π, 1),(,1),(, 1),(,1),(, 1),(,1), Fall, Optimum Demodulation and Detection of CPM Θ =, ± π, ± π, ± π, π π π π π π 3π 3π (, 1),(,1),(, 1),(,1),(, 1),(,1),(, 1)

159 5.3.1 Optimum Demodulation and Detection of CPM Example (count.) If the ytem i in phae tate θ n = -π/4 and I n-1 = -1, then θ = θ n + π h I = π π = π 4 4 n+ 1 n State trelli for partial repone (L = ) CPM with h = 3/4 Fall,

160 5.3.1 Optimum Demodulation and Detection of CPM Example (count.) g(t) i a rectangular pule of duration T with initial tate (,1) A path through the tate trelli correponding to the equence (1, -1, -1, -1, 1, 1) Fall, Phae tree for L = partial repone CPM with h = 3/4

161 5.3.1 Optimum Demodulation and Detection of CPM Metric computation. It i eay to how that the logarithm of the probability of the oberved ignal r(t) conditioned on a particular equence of tranmitted ymbol I i proportional to the cro-correlation metric ( n+ 1) T CM I = r t co ω t + φ t; I dt n () ( ) [ ( )] c ( n+ 1) T = CM n 1() I + r() t co[ ωct + θ ( t; I ) + θ n ] nt The term CM n-1 (I) repreent the metric for the urviving equence up to time nt, and the term ( n 1) T ( ) ( ) [ ( ) ] v + n I; θ n = r t co ω + + nt ct θ t; I θ n dt repreent the additional increment to the metric contributed by the ignal in the time interval nt t (n+1)t. dt Fall,

162 5.3.1 Optimum Demodulation and Detection of CPM There are M L poible equence I = (I n, I n-1,, I n-l+1 ) of ymbol and p (or P) poible phae tate {θ n }. Therefore, there are pm L (or pm L ) different value of v n (I,θ n ) computed in each ignal interval, and each value i ued to increment the metric correponding to the pm L-1 urviving equence from the previou ignaling interval. Computation of metric increment v n (I;θ n ). Fall, 4. 16

163 5.3.1 Optimum Demodulation and Detection of CPM The number of urviving equence at each tate of the Viterbi decoding proce i pm L-1 (or pm L-1 ). For each urviving equence, we have M new increment of v n (I;θ n ) that are added to the exiting metric to yield pm L (or pm L ) equence with pm L (or pm L-1 ) metric. However, thi number i then reduced back to pm L-1 (or pm L-1 ) urvivor with correponding metric by electing the mot probable equence of the M equence merging at each node of the trelli and dicarding the other M-1 equence. Fall,

164 5.3. Performance of CPM Signal In evaluating the performance of CPM ignal achieved with maximum-likelihood equence detection, we mut determine the minimum Euclidean ditance of path through the trelli that eparate that eparate at the node at t = and remerge at a later time at the ame node. Suppoe two ignal i (t) and j (t) correponding to two phae trajectorie φ(t ;I i ) and φ(t ;I j ). The equence I i and I j mut be different in their firt ymbol. Fall,

165 The Euclidean ditance between the two ignal over an interval of length NT (1/T i the ymbol rate) i defined a d ij 5.3. Performance of CPM Signal = = NT NT ( t) j ( t) dt NT NT () t dt + () t dt () t () t [ ] i i ε j NT = Nε co[ ω ( )] [ ( )] ct + φ t; Ii co ωct + φ t; I j dt T ε NT = Nε co[ φ( t; I ) ( )] i φ t; I j dt T ε NT = { 1 co[ φ( t; I ) ( )]} i φ t; I j dt T Hence the Euclidean ditance i related to the phae difference between the path in the tate trelli. i j dt Fall,

166 5.3. Performance of CPM Signal Since ε=ε b log M, may be expreed a d ij dij = where i defined a d ij ε bδ ij log NT { [ ( ) ( )] } M δ ij = 1 co φ t; Ii φ t; I j dt T Furthermore, ince φ(t ;I i )-φ(t ;I j )=φ(t ;I i -I j ), o that, with ξ = I i -I j, may be written a δ ij NT log M δ ij = { 1 coφ( t; ξ )} dt T where any element of ξcan take the value, ±, ±4,, ±(M-1), except that ξ. Fall,

167 5.3. Performance of CPM Signal The error rate performance for CPM i dominated by the term correponding to the minimum Euclidean ditance : εb P M = Kδ Q δ min min N where i the number of path having the minimum ditance K δ min δ min = lim minδ N i, j ij log NT [ ( )] M = lim min 1 coφ t; Ii I j dt N i, j T We note that for conventional binary PSK with no memory, N = 1 and δ = δ. min 1 = Fall,

168 5.3. Performance of CPM Signal Since δ min characterize the performance of PCM, we can invetigate the effect on δ min reulting from varying the alphabet ize M, the modulation index h, and the length of the tranmitted pule in partial repone CPM. Firt, we conider full repone(l = 1) CPM. For M =, we note that the equence Ii = + 1, 1, I, I3 I = 1, + 1, I, I j which differ for k =, 1 and agree for k, reult in two phae trajectorie that merge after the econd ymbol. The correpond to the difference equence ξ = 3 {,,,, } Fall,

169 5.3. Performance of CPM Signal The Euclidean ditance for thi equence i eaily calculated from NT log M δ ij = { 1 coφ( t; ξ )} dt T and provide an upper bound on δ min. Thi upper bound for CPSK with M = i in πh d B ( h) = 1, M = πh Foe example, where h=1/, which correpond to MSK, we have 1 1, o that d B = δ min. For M > and full repone CPM, an upper bound on δ min can be obtained by conidering the phae difference equence ξ= {α, -α,,, } where α=±,±4,,±(m-1). Fall,

170 5.3. Performance of CPM Signal Thi equence ξ = {α, -α,,, } yield the upper bound for M-ary CPFSK a ( ) ( ) in kπh d = B h min log M 1 1 k M 1 kπh Large gain in performance can be achieved by increaing the alphabet ize M. The upper bound may not be achievable for all value of h becaue of δ ( h) d ( h). min B Fall, d B of the modulation index for full repone CPM h

171 Fall, Optimum Receiver For Signal with Random Phae In AWGN Channel In thi ection, we conider the deign of the optimum receiver for carrier modulated ignal when the carrier phae i unknown and no attempt i made to etimate it value. Uncertainty in the carrier phae of the receiver ignal may be due to one or more of the following reaon: The ocillator that are ued at the tranmitter and the receiver to generate the carrier ignal are generally not phae ynchronou. The time delay in the propagation of the ignal from the tranmitter to the receiver i not generally known preciely. Auming a tranmitted ignal of the form ( ) [ ( ) e j π f ] c t = Re g t t that propagate through a channel with delay t will be received a: jπ fc ( t t ) ( ) ( ) ( ) j fct j fc t t Re g t t e π π = = Re g t t e e 171

172 Fall, Optimum Receiver For Signal with Random Phae In AWGN Channel The carrier phae hift due to the propagation delay t i φ = πf c t Note that large change in the carrier phae can occur due to relatively mall change in the propagation delay. For example, if the carrier frequency f c =1 MHz, an uncertainty or a change in the propagation delay of.5μ will caue a phae uncertainty of π rad. In ome channel the time delay in the propagation of the ignal from the tranmitter to the receiver may change rapidly and in an apparently random fahion. In the abence of the knowledge of the carrier phae, we may treat thi ignal parameter a a random variable and determine the form of the optimum receiver for recovering the tranmitted information from the received ignal. 17

173 Fall, Optimum Receiver for Binary Signal We conider a binary communication ytem that ue the two carrier modulated ignal 1 (t) and (t) to tranmit the information, where: jπ fct m() t = Re lm( t) e, m= 1,, t T and lm (t), m=1, are the equivalent low-pa ignal. The two ignal are aumed to have equal energy T T 1 ε = m() t dt = ml () t dt The two ignal are characterized by the complex-valued correlation coefficient 1 T = * ρ1 ρ l1() t l () t dt ε 173

174 5.4.1 Optimum Receiver for Binary Signal The received ignal i aumed to be a phae-hifted verion of the tranmitted ignal and corrupted by the additive noie () [ ( ) ( )] { π } [ ( ) j π f t ] c z t e j f c n t = Re n t c t + jn t e = Re The received ignal may be expreed a r {[ ] φ j π f t } c e ( t) = Re ( t) e j + z( t) jφ where rl t = lm t e + z t t T r l (t) i the equivalent low-pa received ignal. lm () ( ) ( ), Thi received ignal i now paed through a demodulator whoe ampled output at t =T i paed to the detector. Fall,

175 Fall, Optimum Receiver for Binary Signal The optimum demodulator In ection 5.1.1, we demontrated that if the received ignal wa correlated with a et of orthogonal function {f n (t)} that panned the ignal pace, the output from the bank of correlator provide a et of ufficient tatitic for the detector to make a deciion that minimize the probability error. We alo demontrated that a bank of matched filter could be ubtituted for the bank of correlation. A imilar orthogonal decompoition can be employed for a received ignal with an unknown carrier phae. It i mathematically convenient to deal with the equivalent low-pa ignal and to pecify the ignal correlator or matched filter in term of the equivalent low-pa ignal waveform. 175

176 5.4.1 Optimum Receiver for Binary Signal The optimum demodulator The impule repone h l (t) of a filter that i matched to the complex-valued equivalent low-pa ignal l (t), t T i given a (from Problem 5.6) h l (t)= l* (T-t) and the output of T uch filter at t=t i imply (4.1-4) where ε i l () t dt = ε the ignal energy. A imilar reult i obtained if the ignal l (t) i correlated with l* (t) and the correlator i ampled ignal l (t) at t=t. The optimum demodulator for the equivalent low-pa received ignal l (t) may be realized by two matcher filter in parallel, one matched to l1 (t) and the other to l (t). Fall,

177 5.4.1 Optimum Receiver for Binary Signal The optimum demodulator Optimum receiver for binary ignal The output of the matched filter or correlator at the ampling intant are the two complex number r m = r + jr, m =1, mc m Fall,

178 5.4.1 Optimum Receiver for Binary Signal The optimum demodulator Suppoe that the tranmitted ignal i 1 (t). It can be hown that (Problem 5.41) r = ε coφ + n + j ε inφ + n. r 1 = ε ρ co 1c ( 1 ) ( φ α ) + n + j ε ρ in( φ α ) c [ + n ] where ρ i the complex-valued correlation coefficient of two ignal l1 (t) and l (t), which may be expreed a ρ= ρ exp(jα ). The random noie variable n 1c, n 1, n c, and n are jointly Gauian, with zero-mean and equal variance. Fall,

179 5.4.1 Optimum Receiver for Binary Signal The optimum detector The optimum detector oberve the random variable [r 1c r 1 r c r ]=r, where r 1 =r 1c +jr 1, and r =r c +jr, and bae it deciion on the poterior probabilitie P( m r), m=1,. Thee probabilitie may be expreed a P ( r) m ( r m ) P( m ) p() r p =, m = 1, The optimum deciion rule may be expreed a P 1 1 ( ) ( 1 ) > p r r < P( r) or > p( ) < r P ( ) ( ) 1 P 1 Likelihood ratio. Fall,

180 Fall, Optimum Receiver for Binary Signal The optimum detector The ratio of PDF on the left-hand ide i the likelihood ratio, which we denote a p( r 1) Λ () r = p( r ) The right-hand ide i the ratio of the two prior probabilitie, which take the value of unity when the two ignal are equally probable. The probability denity function p(r 1 ) and p(r ) can be obtained by averaging the PDF p(r m,φ) over the PDF of the random carrier phae, i.e., π p( r m) = p( r m, φ) p( φ) dφ 18

181 5.4.1 Optimum Receiver for Binary Signal The optimum detector For the pecial cae in which the two ignal are orthogonal, i.e., ρ=, the output of the demodulator are (from 5.4-1): r1 = r1 c + jr1 = εco φ + n1 c + j(εin φ + n1) r = r + jr c = n + jn c where (n 1c, n 1, n c, n ) are mutually uncorrelated and, tatitically independent, zero-mean Gauian random variable. Fall,

182 5.4.1 Optimum Receiver for Binary Signal The optimum detector The joint PDF of r=[r 1c r 1 r c r ] may be expreed a a product of the marginal PDF: 1 ( r1 c εco φ) + ( r1 εin φ) pr ( 1c, r1 1, φ) = exp πσ σ 1 rc + r pr ( c, r 1, φ) = exp πσ σ where σ =εn. The uniform PDF for the carrier phae φ (p(φ)=1/π) repreent the mot ignorance that can be exhibited by the detector. Thi i called the leat favorable PDF for φ. Fall, 4. 18

183 5.4.1 Optimum Receiver for Binary Signal The optimum detector With p(φ)=1/π, φ π, ubtituted into the integral in p(r m ), we obtain: 1 π p( r1 1 1 ) c, r, φ dφ π 1 r ( 1c co 1in c + r r r c + ε π ε φ + φ) = exp exp dφ πσ σ π σ ( ) 1 π ε r1 ccoφ + r1inφ ε r1 c + r 1 exp dφ = I π σ σ where I (x) i the modified Beel function of zeroth order. Fall,

184 Fall, Optimum Receiver for Binary Signal The optimum detector By performing a imilar integration under the aumption that the ignal (t) wa tranmitted, we obtain the reult 1 r c + r + 4ε ε rc + r p( rc, r ) = exp I πσ σ σ When we ubtitute thee reult into the likelihood ratio given by equation Λ(r), we obtain the reult ( ) I 1 ε r 1c + r1 σ P( ) Λ ( ) = ><? r I ε r + r σ P ( 1 ) ( ) c The optimum detector compute the two envelope r and 1c + r r 1 c + r and the correponding value of the Beel function ( ) I ε r 1 c + r1 / σ and I ε r + r σ to form the likelihood ratio. ( ) c / 184

185 Fall, Optimum Receiver for Binary Signal The optimum detector we oberve that thi computation require knowledge of the noie variance σ. The likelihood ratio i then compared with the threhold P( )/P( 1 ) to determine which ignal wa tranmitted. A ignificant implification in the implementation of the optimum detector occur when the two ignal are equally probable. In uch a cae the threhold become unity, and, due to the monotonicity of Beel function hown in figure, the optimum detection rule implifie to 1 1c + > 1 < c + r r r r 185

186 5.4.1 Optimum Receiver for Binary Signal The optimum detector The optimum detector bae it deciion on the two envelope and r, and, it i called an envelope detector. 1c + r r 1 c + r We oberve that the computation of the envelope of the received ignal ample at the output of the demodulator render the carrier phae irrelevant in the deciion a to which ignal wa tranmitted. Equivalently, the deciion may be baed on the computation of the quared envelope r 1c +r 1 and r c +r, in which cae the detector i call a quare-law detector. Fall,

187 Fall, Optimum Receiver for Binary Signal Detection of binary FSK ignal Recall that in binary FSK we employ two different frequencie, ay f 1 and f =f 1 +Δf, to tranmit a binary information equence. The choe of minimum frequency eparation Δf = f - f 1 i conidered below: ( ) ( ) t = ε T co π ft, t = ε T co π f t, t T 1 b b 1 b b b The equivalent low-pa counterpart are: j () ε ( ) ε t = T, t = T e, t T π ft l1 b b l b b b The received ignal may be expreed a: ε b r() t = co( π fmt+ φm) + n( t), m=,1 T b 187

188 5.4.1 Optimum Receiver for Binary Signal Detection of binary FSK ignal Demodulation and quare-law detection: fkc () t = co ( π f1 + πk f ) t, k =,1 T b Fall, 4. fk () t = in ( π f1 + πk f ) t, k =,1 T b 188

189 Tb Tb ε b rkc = r() t if () co ( 1 ) ( ) co ( 1 ) kc t dt = π f + m f t+ φ m + n t π f + πk f t dt T i b T b ε T b b = { co π ( f1 m f ) t coφ in π ( f1 m f ) t inφ }{ co π ( f1 k f ) t } dt n T i ε b Optimum Receiver for Binary Signal Detection of binary FSK ignal If the mth ignal i tranmitted, the four ample at the detector may be expreed a: { co π ( m k) ft co π ( f1 ( m k) f ) t } T b b = + + T b { ( ) ( 1 ( ) ) } ( ) ( ) ( ) ( ) m m kc ( ) ( ) ( ) ( ) + coφm in π m k ft in π f + m+ k f t inφmdt+ n b ε in co b π m k ft π m k ft = coφ + inφ + n Tb π m k f π m k f m m kc in π m k ftb co π m k ε = b coφm + ftb 1 in φm + nkc k, m=,1 π m k ftb π m k ftb Fall, T kc

190 5.4.1 Optimum Receiver for Binary Signal Detection of binary FSK ignal If the mth ignal i tranmitted, the four ample at the detector may be expreed a (cont.): Tb Tb ε b rk = r() t if () co ( 1 ) ( ) in ( 1 ) k t dt = π f + m f t+ φ m + n t π f + πk f t dt T i b T b ε T b b = { co π ( f1 m f ) t coφ in π ( f1 m f ) t inφ }{ in π ( f1 k f ) t } dt n T i ε b { in π ( m k) ft in π ( f1 ( m k) f ) t } T b b = + + T b { ( ) ( 1 ( ) ) } ( ) ( ) ( ) ( ) m m k ( ) ( ) + coφm co π m k ft co π f + m+ k f t inφmdt+ n b ε co in b π m k ft π m k ft = coφ inφ + n Tb π m k f π m k f m m k ( ) co π m k ftb 1 in π m ε = b coφm k ftb in φm + nk k, m=,1 π m k ftb π ( m k) ftb Fall, T k

191 5.4.1 Optimum Receiver for Binary Signal Detection of binary FSK ignal We oberve that when k=m, the ampled value to the detector are rmc = εb coφm + nmc k=m r = ε inφ + n m b m m We oberve that when k m, the ignal component in the ample r kc and r k will vanih, independently of the value of the phae hift φ k, provided that the frequency eparation between ucceive frequency i Δf= 1/T. In uch cae, the other two correlator output conit of noie only, i.e., r kc =n kc, r k =n k, k m. Fall,

192 Fall, Optimum Receiver for M-ary Orthogonal Signal If the equal energy and equally probable ignal waveform are repreented a jπ fct m( t) = Re lm( t) e, m= 1,,..., M, t T where lm (t) are the equivalent low-pa ignal. The optimum correlation-type or matched-filter-type demodulator produce the M complex-valued random variable T * r = r + jr = r t h T t dt () ( ) () ( ) T ( ) () () m mc m l lm T * * = r l t lm T T t dt = r l t lm t dt, m= 1,,..., M where r l (t) i the equivalent low-pa ignal. The optimum detector, baed on a random, uniformly ditributed carrier phae, compute the M envelope r m = rmc + rm, m= 1,,..., M or, the quared envelope r m, and elect the ignal with the larget envelope. 19

193 5.4. Optimum Receiver for M-ary Orthogonal Signal Optimum receiver for M-ary orthogonal FSK ignal. There are M correlator: two for each poible tranmitted frequency. The minimum frequency eparation between adjacent frequencie to maintain orthogonality i Δf= 1/T. Fall, Demodulation of M-ary ignal for noncoherent detection

194 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal We aume that the M ignal are equally probable a priori and that the ignal 1 (t) i tranmitted in the ignal interval t T. The M deciion metric at the detector are the M envelope where r r = r + r, m= 1,,..., M m mc m = ε coφ + n 1c 1 1c r = ε inφ + n and rmc = nmc, rm = nm, m=,3,..., M The additive noie component {n mc } and {n m } are mutually tatitically independent zero-mean Gauian variable with equal variance σ =N /. Fall,

195 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal The PDF of the random variable at the input to the detector are ( ) 1 r1 c r ε 1 r1 c r + + ε + 1 p ( r ) 1 1c, r1 exp I r = πσ σ σ 1 rmc + r m pr ( r, ) exp,,3,..., m mc rm = m M = πσ σ We define the normalized variable R m Θ = tan m = r + r σ r r mc m 1 m mc Fall,

196 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal Clearly, r mc =σr m coθ m and r m =σr m inθ m. The Jacobian of thi tranformation i σ co Θm σ in Θm J = = σ Rm σrmin Θm σrmco Θm Conequently, R 1 1 ε ε pr ( 1, Θ 1) = exp R1 + I R1 π N N Rm 1 p ( Rm, Θ m) = exp Rm, m=,3,..., M π Finally, by averaging p(r m,θ m ) over Θ m, the factor of π i eliminated. Thu, we find that R 1 ha a Rice probability ditribution and R m, m=,3,,m, are each Rayleigh-ditribued. Fall,

197 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal The probability of a correct deciion i imply the probability that R 1 >R, and R 1 >R 3,,and R 1 >R m. Hence, = P R < R1, R3 < R1,, R M < R1 R1 = x pr x dx 1 Becaue the random variable R m, m=,3,,m, are tatitically independent and identically ditributed, the joint probability conditioned on R 1 factor into a product of M-1 identical term. where c (,,, ) P = P R < R R < R R < R M 1 ( ) ( ) M 1 ( ) ( ) Pc = P R < R1 R1 = x pr x dx 1 x ( ) ( ) P R R R x p r dr e x / < 1 1 = = R = 1 (From Eq.1-19) (5.4-4) Fall,

198 The (M-1)th power may be expreed a ( ) M 1 M 1 x / n M 1 nx 1 e = ( 1) e n= n Subtitution of thi reult into Equation and integration over x yield the probability of a correct deciion a M 1 n M 1 1 nε Pc = ( 1) exp n= n n+ 1 ( n+ 1) N where ε /N i the SNR per ymbol. The probability of a ymbol error, which i P m =1-P c, become M 1 n+ 1 M 1 1 nkε b PM = ( 1) exp n= 1 n n+ 1 ( n+ 1) N where ε b /N i the SNR per bit. Fall, Probability of Error for Envelope Detection of M-ary Orthogonal Signal 198

199 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal For binary orthogonal ignal (M=), Equation reduce to the imple form 1 P e ε b N = For M>, we may compute the probability of a bit error by making ue of the relationhip P b = k 1 P k 1 which wa etablihed in Section 5.. Figure how the bit-error probability a a function of SNR per bit γ b for M=,4,8,16, and 3. M Fall,

200 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal Jut a in the cae of coherent detection of M-ary orthogonal ignal, we oberve that for any given bit-error probability, the SNR per bit decreae a M increae. It will be how in Chapter 7 that, in the limit a M, the probability of bit error P b can be made arbitrarily mall provided that the SNR per bit i greater than the Shannon limit of -1.6dB. The cot for increaing M i the bandwidth required to tranmit the ignal. For M-ary FSK, the frequency eparation between adjacent frequencie i Δf=1/T for ignal orthogonality. The bandwidth required for the M ignal i W=M Δf=M/T. Fall, 4.

201 5.4.3 Probability of Error for Envelope Detection of M-ary Orthogonal Signal The bit rate i R=k/T, where k=log M. Therefore, the bit rate-to-bandwidth ratio i R W = log M M Fall, 4. 1

202 5.4.4 Probability of Error for Envelope Detection of Correlated Binary Signal In thi ection, we conider the performance of the envelope detector for binary, equal-energy correlated ignal. When the two ignal are correlated, the input to the detector i the complex-valued random variable given by Equation We aume that the detector bae it deciion on the envelope r 1 and r, which are correlated (tatitically dependent). The marginal PDF of R 1 = r 1 and R = r are Ricean ditributed and may be expreed a R m Rm + β m βmr m exp I ( Rm > ) pr ( ) 4 m = εn εn εn ( Rm < ) m=1,, where β 1 =ε and β =ε ρ, baed on the aumption that ignal 1 (t) wa tranmitted. Fall, 4.

203 Fall, Probability of Error for Envelope Detection of Correlated Binary Signal Since R 1 and R are tatitically dependent a a conequence of the nonorthogonality of the ignal, the probability of error may be obtained by evaluating the double integral b ( ) (, ) P P R R p x x dx dx = > = 1 x 1 1 where p(x 1,x ) i the joint PDF of the envelope R 1 and R. Thi approach wa firt ued by Heltrom (1995), who determined the joint PDF of R 1 and R and evaluated the double integral. An alternative approach i baed on the obervation that the probability of error may alo be expreed a b 3 1 ( ) ( ) ( ) P = P R > R = P R > R = P R R >

204 5.4.4 Probability of Error for Envelope Detection of Correlated Binary Signal But R -R 1 i a pecial cae of a general quadratic form in complex-valued Gauian random variable, treated later in Appendix B. For the pecial cae under conideration, the derivation yield the error probability in the form where b 1 ( ) ( a + b, ) ( ) P = Q a b e I ab 1 ( ) ε ( ) 1 1 ρ b 1 1 ρ εb a= = + N N b Q 1 (a, b) i the Marcum Q function defined in.1-13 and I (x) i the modified Beel function of order zero. Fall, 4. 4

205 5.4.4 Probability of Error for Envelope Detection of Correlated Binary Signal The error probability P b i illutrated in Figure for everal value of ρ. Probability of error for noncoherent detection of binary FSK P b i minimized when ρ=; that i, when the ignal are orthogonal. Fall, 4. 5